Jump Your Steps: Optimizing Sampling Schedule of Discrete Diffusion Models

## Abstract

Abstract Diffusion models have seen notable success in continuous domains, leading to the development of discrete diffusion models (DDMs) for discrete variables. Despite recent advances, DDMs face the challenge of slow sampling speeds. While parallel sampling methods like τ 𝜏 \tau italic_τ -leaping accelerate this process, they introduce Compounding Decoding Error (CDE), where discrepancies arise between the true distribution and the approximation from parallel token generation, leading to degraded sample quality. In this work, we present Jump Your Steps (JYS), a novel approach that optimizes the allocation of discrete sampling timesteps by minimizing CDE without extra computational cost. More precisely, we derive a practical upper bound on CDE and propose an efficient algorithm for searching for the optimal sampling schedule. Extensive experiments across image, music, and text generation show that JYS significantly improves sampling quality, establishing it as a versatile framework for enhancing DDM performance for fast sampling.

### 1 Introduction

Diffusion models (Sohl-Dickstein et al., ; Song et al., ; Ho et al., ; Song et al., ; Karras et al., ) have achieved remarkable success in generation tasks within the continuous domain. However, certain modalities, such as text and music, inherently possess discrete features. Recently, discrete diffusion models (DDMs) (Austin et al., ; Campbell et al., ; ; Gat et al., ) have demonstrated performance comparable to state-of-the-art methods in various areas, including text (Lou et al., ; Shi et al., ) and image (Chang et al., ; Gu et al., ) generation. Nevertheless, like their continuous counterparts, DDMs encounter a significant bottleneck in sampling speed due to their progressive refinement process.

In contrast to continuous-domain diffusion models, where sampling dynamics are driven by sample-wise differential equations (Song et al., ), allowing for the direct application of well-established numerical methods to accelerate generation, enhancing speed in DDMs poses a significant challenge. To address this, researchers have proposed fast and efficient samplers, including notable methods such as the $\tau$-leaping (Campbell et al., ; Lezama et al., ; Sun et al., ) and $k$-Gillespie algorithms (Zhao et al., ), which facilitate parallel sampling of multiple tokens in a single step. However, this parallel but independent sampling introduces Compounding Decoding Error (CDE) (Lezama et al., ), which arises from a mismatch between the training and inference distributions of intermediate latents during parallel sampling. Specifically, while each token is generated according to its marginal distribution, the joint distribution deviates from the learned distribution. To mitigate this issue, the predictor-corrector (PC) sampler (Campbell et al., ) has been proposed. This sampler slightly perturbs the generated data to correct incorrectly generated tokens. However, these methods have limitations, including impracticality under low computational budgets (Campbell et al., ), the need for an additional corrector (Lezama et al., ), or reliance on specialized architectures and loss functions (Zhao et al., ).

Figure: Figure 1: (Top) Comparison of sampling trajectories: ground truth vs. parallel sampling using a uniform schedule and the Jump Your Steps (JYS) schedule. (Bottom) Uniform schedule exhibits compounding decoding errors during parallel sampling, while JYS reduces them by using fewer steps in deterministic phases and reallocating skipped steps to other timesteps.
Refer to caption: x1.png

To reduce CDE and enable fast sampling in DDM fundamentally, we first introduce a rigorous quantity to measure CDE (see Figure 1 Top, and Section 3.1) and propose a novel approach called Jump Your Steps (JYS), which optimizes the allocation of discrete sampling timesteps $\{T\mathrel{\mathchoice{\mkern 2.0mu\leavevmode\leavevmode\hbox{ \leavevmode
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Unlike previous methods such as the PC sampler, our approach requires no additional computational resources or modifications to the model architecture or loss function. We empirically validate the effectiveness of our sampling schedule across various datasets, including synthetic sequential data, CIFAR-10 (image), Lakh Pianoroll (music), and text modeling. Our approach accelerates DDM sampling across models using different forward corruption transition kernels, such as uniform, Gaussian, and absorbing transition matrices. Our comprehensive experiments cover both unconditional and conditional generation tasks, consistently showing that optimizing the sampling schedule significantly enhances sampling quality. These results indicate that our method serves as a general framework for speeding up discrete diffusion model sampling.

### 2 Background

#### 2.1 Continuous Time Framework for Discrete Diffusion Models.

Discrete diffusion models (DDMs) define the generative process as the reverse of the data-corrupting forward process, expressed as a Continuous Time Markov Chain (CTMC) on a finite state space $\mathcal{S}$ (Campbell et al., ). For the data-corrupting process $(X_{t})_{t\in[0,T]}$, the density evolution is described as:

$$ $q_{t+dt|t}(y\mid x)=\delta_{xy}+R_{t}(x,y)dt+o(dt)$ (1) $$

Here, $\delta_{xy}$ is the Dirac delta function, $R_{t}\in\mathbb{R}^{S\times S}$ is the transition rate matrix of the forward CTMC, with $S=|\mathcal{S}|$, and $dt>0$. Rate matrices ensure the marginal distribution $q_{t}(x_{t})=\int q_{t}(x_{t}|x_{0})q_{0}(x_{0})dx_{0}$, where $q_{0}=p_{\mathrm{data}}$ and $q_{T}\approx\pi$, the stationary distribution of the forward CTMC. Various transition matrices have been proposed, allowing $\pi$ to follow a uniform or Gaussian distribution, or converting samples into masked tokens.

For generation, we reverse the forward process, moving from the marginal $q_{T}$ back to $p_{\mathrm{data}}$. This time-reversal CTMC is also a CTMC (Campbell et al., ; ):

$$ $q_{t-dt|t}(y\mid x)=\delta_{xy}+\tilde{R}(x,y)dt+o(dt),$ (2) $$

where the backward transition rate $\tilde{R}$ is defined as:

$$ $\tilde{R}(x,y)=R(y,x)\underbrace{\frac{q_{t}(y)}{q_{t}(x)}}_{\mathclap{\textrm {Score Parametrization}}}=R(y,x)\sum_{x_{0}}\frac{q_{t}(y\mid x_{0})}{q_{t}(x \mid x_{0})}\underbrace{q_{0|t}(x_{0}\mid x)}_{\mathclap{\textrm{Denoising Parametrization}}}$ $$

The literature primarily falls into two parameterizations: Denoising parameterization (Campbell et al., ; Austin et al., ; Campbell et al., ) approximates a parameterized denoising model as $p^{\theta}_{0|t}(x_{0}|x)\approx q_{0|t}(x_{0}|x)$. Conversely, score parameterization (Lou et al., ; Meng et al., ) estimates the ratio of the data distribution as $s_{t}^{\theta}(y|x)={q_{t}(y)}/{q_{t}(x)}$.

#### 2.2 Sampling from the backward CTMC

##### Gillespie’s Algorithm

was proposed as a simulation algorithm for a given CTMC (Gillespie, ). Gillespie’s algorithm simulates the CTMC by calculating the rate matrix at each state transition. If the rate matrix of the CTMC depends only on the state, Gillespie’s algorithm serves as an exact simulation method. However, since it allows for only one token transition each time the rate matrix is calculated, it is computationally inefficient.

##### k 𝑘 k italic_k -Gillespie’s Algorithm

Instead of updating only one token for each rate matrix calculation, the $k$-Gillespie’s algorithm (Zhao et al., ) updates $k$ tokens in parallel. This reduces the computation by a factor of $1/k$ compared to the original Gillespie algorithm.

##### τ 𝜏 \tau italic_τ -Leaping

On the other hand, Campbell et al. () proposes sampling through $\tau$-leaping. Unlike the $k$-Gillespie algorithm, which update $k$ tokens in parallel, $\tau$-leaping simultaneously updates all tokens according to the given fixed rate matrix within the specified time interval $[t,t+\tau)$. Recently, Tweedie $\tau$-leaping, which considers changes in the rate matrix according to the noise schedule, has been proposed (Sun et al., ; Lou et al., ).

### 3 Optimizing the sampling schedule of discrete diffusion models

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Figure 2 summarizes the relationships between the theoretical analyses discussed in this section.

Although this section focuses on samplers based on $\tau$-leaping, all methods are also applicable to the $k$-Gillespie algorithm. For extensions to $k$-Gillespie, please refer to the Algorithm 1.

##### Notations

To begin, we introduce some essential mathematical notation. $X:$ a random variable, $\mathbf{x}:$ its observation, $\mathbb{P},\mathbb{Q}:$ distributions, $\{T\mathrel{\mathchoice{\mkern 2.0mu\leavevmode\leavevmode\hbox{ \leavevmode
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to
0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.41666pt}\pgfsys@lineto{0.0pt}{3.57222pt
}\pgfsys@lineto{3.19446pt}{3.57222pt}\pgfsys@lineto{3.19446pt}{-0.41666pt}
\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.19444pt\hbox
{\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}c\}$. For clarity, when working with backward CTMCs, we slightly abuse notation and express intervals as $[s,t]\triangleq\{u\mid s\geq u\geq t\}$;
the same applies to open and half-open intervals.

Figure: Figure 2: An illustration of the relationship between the KL divergence of the distribution, the compounding error $\mathcal{E}_{\mathrm{CDE}}$ (Section 3.1), and KLUB (Section 3.3). The sampling schedule $\{T\mathrel{\mathchoice{\mkern 2.0mu\leavevmode\leavevmode\hbox{ \leavevmode \hbox to6.39pt{\vbox to7.98pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower -0.83333pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{ pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ } \pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt} \pgfsys@invoke{ }\nullfont\hbox to0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.83333 pt}\pgfsys@lineto{0.0pt}{7.14444pt}\pgfsys@lineto{6.38892pt}{7.14444pt} \pgfsys@lineto{6.38892pt}{-0.83333pt}\pgfsys@closepath\pgfsys@clipnext \pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}}{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-6.3889pt\hbox{ \set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to6.39pt{\vbox to7.98pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.83333pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.83333pt}\pgfsys@lineto{0.0pt}{7.14444pt }\pgfsys@lineto{6.38892pt}{7.14444pt}\pgfsys@lineto{6.38892pt}{-0.83333pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-6.3889pt\hbox{ \set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.19pt{\vbox to3.99pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.41666pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.41666pt}\pgfsys@lineto{0.0pt}{3.57222pt }\pgfsys@lineto{3.19446pt}{3.57222pt}\pgfsys@lineto{3.19446pt}{-0.41666pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.19444pt\hbox {\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}t_{1}\mathrel{ \mathchoice{\mkern 2.0mu\leavevmode\leavevmode\hbox{ \leavevmode\hbox to6.39pt {\vbox to7.98pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.83333pt \hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{ rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ } \pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt} \pgfsys@invoke{ }\nullfont\hbox to0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.83333 pt}\pgfsys@lineto{0.0pt}{7.14444pt}\pgfsys@lineto{6.38892pt}{7.14444pt} \pgfsys@lineto{6.38892pt}{-0.83333pt}\pgfsys@closepath\pgfsys@clipnext \pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}}{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-6.3889pt\hbox{ \set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to6.39pt{\vbox to7.98pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.83333pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.83333pt}\pgfsys@lineto{0.0pt}{7.14444pt }\pgfsys@lineto{6.38892pt}{7.14444pt}\pgfsys@lineto{6.38892pt}{-0.83333pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-6.3889pt\hbox{ \set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.19pt{\vbox to3.99pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.41666pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.41666pt}\pgfsys@lineto{0.0pt}{3.57222pt }\pgfsys@lineto{3.19446pt}{3.57222pt}\pgfsys@lineto{3.19446pt}{-0.41666pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.19444pt\hbox {\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}\dots\mathrel{ \mathchoice{\mkern 2.0mu\leavevmode\leavevmode\hbox{ \leavevmode\hbox to6.39pt {\vbox to7.98pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.83333pt \hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{ rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ } \pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt} \pgfsys@invoke{ }\nullfont\hbox to0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.83333 pt}\pgfsys@lineto{0.0pt}{7.14444pt}\pgfsys@lineto{6.38892pt}{7.14444pt} \pgfsys@lineto{6.38892pt}{-0.83333pt}\pgfsys@closepath\pgfsys@clipnext \pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}}{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-6.3889pt\hbox{ \set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to6.39pt{\vbox to7.98pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.83333pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.83333pt}\pgfsys@lineto{0.0pt}{7.14444pt }\pgfsys@lineto{6.38892pt}{7.14444pt}\pgfsys@lineto{6.38892pt}{-0.83333pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-6.3889pt\hbox{ \set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.19pt{\vbox to3.99pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.41666pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.41666pt}\pgfsys@lineto{0.0pt}{3.57222pt }\pgfsys@lineto{3.19446pt}{3.57222pt}\pgfsys@lineto{3.19446pt}{-0.41666pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.19444pt\hbox {\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}t_{N-1}\mathrel{ \mathchoice{\mkern 2.0mu\leavevmode\leavevmode\hbox{ \leavevmode\hbox to6.39pt {\vbox to7.98pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.83333pt \hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{ rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ } \pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt} \pgfsys@invoke{ }\nullfont\hbox to0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.83333 pt}\pgfsys@lineto{0.0pt}{7.14444pt}\pgfsys@lineto{6.38892pt}{7.14444pt} \pgfsys@lineto{6.38892pt}{-0.83333pt}\pgfsys@closepath\pgfsys@clipnext \pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}}{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-6.3889pt\hbox{ \set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to6.39pt{\vbox to7.98pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.83333pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.83333pt}\pgfsys@lineto{0.0pt}{7.14444pt }\pgfsys@lineto{6.38892pt}{7.14444pt}\pgfsys@lineto{6.38892pt}{-0.83333pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-6.3889pt\hbox{ \set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.19pt{\vbox to3.99pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.41666pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.41666pt}\pgfsys@lineto{0.0pt}{3.57222pt }\pgfsys@lineto{3.19446pt}{3.57222pt}\pgfsys@lineto{3.19446pt}{-0.41666pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.19444pt\hbox {\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}0\}$ is optimized to minimize KLUB using the efficient algorithms detailed in Section 3.4, and 3.5.

#### 3.1 Time-Dependent Nature of Compounding Decoding Errors

We introduce a measure for the *Compounding Decoding Error* (CDE), $\mathcal{E}_{\mathrm{CDE}}$, which quantifies the discrepancy between the true joint distribution and the distribution from parallel token generation. For illustration, we consider a discrete process $X_{t}=(X^{1}_{t},X^{2}_{t})$ with sequence length 2, consisting of tokens $X^{1}_{t}$ and $X^{2}_{t}$. The general case is provided in Appendix A.1.

We propose measuring the CDE for a single parallel sampling step $\{s\mathrel{\mathchoice{\mkern 2.0mu\leavevmode\leavevmode\hbox{ \leavevmode
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\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
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{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
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{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
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{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
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\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
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{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
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{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
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$$ $\mathcal{E}_{\mathrm{CDE}}(s\shortrightarrow t|\mathbf{x}_{s})\triangleq \mathcal{D}_{\mathrm{KL}}(\underbrace{P_{X^{1}_{t},X^{2}_{t}|\mathbf{x}_{s}}}_ {\textrm{True distribution}}\|\underbrace{P_{X^{1}_{t}|\mathbf{x}_{s}}\otimes P _{X^{2}_{t}|\mathbf{x}_{s}}}_{\textrm{Approx.\ distribution from parallel sampling}}).$ (3) $$

We note that the defined CDE is equivalent to the conditional mutual information $\mathcal{I}(X^{1}_{t};X^{2}_{t}|\mathbf{x}_{s})$ of tokens $X^{1}_{t},X^{2}_{t}$:

$$ $\mathcal{E}_{\mathrm{CDE}}(s\shortrightarrow t|\mathbf{x}_{s})=\mathcal{I}(X^{ 1}_{t};X^{2}_{t}|\mathbf{x}_{s}).$ (4) $$

This expression links the compounding error to the mutual information between tokens; lower mutual information reduces parallel sampling errors. For example, as shown in Figure 1 (Bottom), as generation progresses, the uncertainty of each token decreases over time due to the tokens already generated, reducing mutual information and preventing CDE. In general, CDE depends on the timesteps, and its behavior varies with the data distribution, corruption kernel (see Fig. 9 for illustration), and DDM sampling methods. Motivated by this observation, we hypothesize that we can reduce the CDEs during generation process by optimizing the sampling schedule.

#### 3.2 Relation between Compounding Decoding errors and generation quality

While the Eq. (3) allows us to estimate the CDE starting from a specific state $\mathbf{x}_{s}$, in practice, we are interested in the average compounding error over all possible starting states at time $s$. To assess the overall impact of the CDE when transitioning over the timesteps $s\shortrightarrow t$, we consider the expected value of $\mathcal{E}_{\mathrm{CDE}}(s\shortrightarrow t|\mathbf{x}_{s})$ with respect to $\mathbf{x}_{s}\sim\mathbb{P}_{s}$. This leads us to consider:

$$ $\mathcal{E}_{\mathrm{CDE}}(s\shortrightarrow t)\triangleq\mathbb{E}_{\mathbf{x }_{s}}\left[\mathcal{E}_{\mathrm{CDE}}(s\shortrightarrow t|\mathbf{x}_{s}) \right].$ (5) $$

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$$ $\min_{t_{1},t_{2},\dots,t_{N-1}}\sum_{i=0}^{N-1}\mathcal{E}_{\mathrm{CDE}}(t_{ i}\shortrightarrow t_{i+1}).$ (6) $$

Interestingly, we find in the following theorem that cumulative CDEs over the sampling schedule can upper bound the KL divergence between the true distribution at time $t=0$, denoted $\mathbb{P}_{0}$, and the distribution $\mathbb{Q}_{0}^{T\mathrel{\mathchoice{\mkern 2.0mu\leavevmode\leavevmode\hbox{
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{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox
{\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
\leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.13pt{\vbox to3.91pt{
\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.40833pt\hbox to0.0pt{
\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}
{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to
0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.40833pt}\pgfsys@lineto{0.0pt}{3.50076pt
}\pgfsys@lineto{3.13055pt}{3.50076pt}\pgfsys@lineto{3.13055pt}{-0.40833pt}
\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.13055pt\hbox
{\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
\leavevmode\leavevmode\hbox{ \leavevmode\hbox to2.24pt{\vbox to2.79pt{
\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{
\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}
{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to
0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt
}\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt}
\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox
{\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}0}$ obtained from parallel sampling along the sampling schedule (see Appendix A.1 for proof):

###### Theorem 3.1 .

We have the following bound on the KL divergence between $\mathbb{P}_{0}$ and $\mathbb{Q}_{0}^{T\mathrel{\mathchoice{\mkern 2.0mu\leavevmode\leavevmode\hbox{
\leavevmode\hbox to4.47pt{\vbox to5.58pt{\pgfpicture\makeatletter\hbox{\hskip
0
.0pt\lower-0.58333pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}
\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }
\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{{}{}{}{}
\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt}\pgfsys@lineto
{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt}\pgfsys@closepath
\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox
{\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
\leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{
\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{
\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}
{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to
0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt}
\pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt}
\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox
{\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
\leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.13pt{\vbox to3.91pt{
\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.40833pt\hbox to0.0pt{
\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}
{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to
0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.40833pt}\pgfsys@lineto{0.0pt}{3.50076pt
}\pgfsys@lineto{3.13055pt}{3.50076pt}\pgfsys@lineto{3.13055pt}{-0.40833pt}
\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.13055pt\hbox
{\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
\leavevmode\leavevmode\hbox{ \leavevmode\hbox to2.24pt{\vbox to2.79pt{
\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{
\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}
{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to
0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt
}\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt}
\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox
{\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}t_{1}\mathrel{
\mathchoice{\mkern 2.0mu\leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt
{\vbox to5.58pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt
\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{
rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }
\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}
\pgfsys@invoke{ }\nullfont\hbox to0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333
pt}\pgfsys@lineto{0.0pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{5.0011pt}
\pgfsys@lineto{4.47223pt}{-0.58333pt}\pgfsys@closepath\pgfsys@clipnext
\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox
{\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
\leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{
\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{
\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}
{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to
0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt}
\pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt}
\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox
{\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
\leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.13pt{\vbox to3.91pt{
\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.40833pt\hbox to0.0pt{
\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}
{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to
0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.40833pt}\pgfsys@lineto{0.0pt}{3.50076pt
}\pgfsys@lineto{3.13055pt}{3.50076pt}\pgfsys@lineto{3.13055pt}{-0.40833pt}
\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.13055pt\hbox
{\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
\leavevmode\leavevmode\hbox{ \leavevmode\hbox to2.24pt{\vbox to2.79pt{
\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{
\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}
{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to
0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt
}\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt}
\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox
{\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}\cdots\mathrel{
\mathchoice{\mkern 2.0mu\leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt
{\vbox to5.58pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt
\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{
rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }
\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}
\pgfsys@invoke{ }\nullfont\hbox to0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333
pt}\pgfsys@lineto{0.0pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{5.0011pt}
\pgfsys@lineto{4.47223pt}{-0.58333pt}\pgfsys@closepath\pgfsys@clipnext
\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox
{\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
\leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{
\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{
\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}
{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to
0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt}
\pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt}
\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox
{\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
\leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.13pt{\vbox to3.91pt{
\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.40833pt\hbox to0.0pt{
\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}
{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to
0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.40833pt}\pgfsys@lineto{0.0pt}{3.50076pt
}\pgfsys@lineto{3.13055pt}{3.50076pt}\pgfsys@lineto{3.13055pt}{-0.40833pt}
\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.13055pt\hbox
{\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
\leavevmode\leavevmode\hbox{ \leavevmode\hbox to2.24pt{\vbox to2.79pt{
\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{
\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}
{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to
0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt
}\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt}
\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox
{\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}0}$ in terms of cumulative CDEs:

$$ $\mathcal{D}_{\mathrm{KL}}(\mathbb{P}_{0}\|\mathbb{Q}_{0}^{T\mathrel{ \mathchoice{\mkern 2.0mu\leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt {\vbox to5.58pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt \hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{ rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ } \pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt} \pgfsys@invoke{ }\nullfont\hbox to0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333 pt}\pgfsys@lineto{0.0pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{-0.58333pt}\pgfsys@closepath\pgfsys@clipnext \pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}}{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.13pt{\vbox to3.91pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.40833pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.40833pt}\pgfsys@lineto{0.0pt}{3.50076pt }\pgfsys@lineto{3.13055pt}{3.50076pt}\pgfsys@lineto{3.13055pt}{-0.40833pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.13055pt\hbox {\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to2.24pt{\vbox to2.79pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt }\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox {\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}t_{1}\mathrel{ \mathchoice{\mkern 2.0mu\leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt {\vbox to5.58pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt \hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{ rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ } \pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt} \pgfsys@invoke{ }\nullfont\hbox to0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333 pt}\pgfsys@lineto{0.0pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{-0.58333pt}\pgfsys@closepath\pgfsys@clipnext \pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}}{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.13pt{\vbox to3.91pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.40833pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.40833pt}\pgfsys@lineto{0.0pt}{3.50076pt }\pgfsys@lineto{3.13055pt}{3.50076pt}\pgfsys@lineto{3.13055pt}{-0.40833pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.13055pt\hbox {\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to2.24pt{\vbox to2.79pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt }\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox {\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}\cdots\mathrel{ \mathchoice{\mkern 2.0mu\leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt {\vbox to5.58pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt \hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{ rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ } \pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt} \pgfsys@invoke{ }\nullfont\hbox to0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333 pt}\pgfsys@lineto{0.0pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{-0.58333pt}\pgfsys@closepath\pgfsys@clipnext \pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}}{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.13pt{\vbox to3.91pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.40833pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.40833pt}\pgfsys@lineto{0.0pt}{3.50076pt }\pgfsys@lineto{3.13055pt}{3.50076pt}\pgfsys@lineto{3.13055pt}{-0.40833pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.13055pt\hbox {\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to2.24pt{\vbox to2.79pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt }\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox {\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}0})\leq\sum_{i=0}^{N -1}\mathcal{E}_{\mathrm{CDE}}(t_{i}\shortrightarrow t_{i+1}).$ (7) $$

This theorem suggests that effectively allocating the time schedule to minimize the CDEs can implicitly reduce the discrepancy between the true distribution and the approximate distribution obtained from parallel sampling. Motivated by this, in the next section, we derive a tractable upper bound for
$\sum_{i=0}^{N-1}\mathcal{E}_{\mathrm{CDE}}(t_{i}\shortrightarrow t_{i+1})$ that depends on the sampling schedule $\{T\mathrel{\mathchoice{\mkern 2.0mu\leavevmode\leavevmode\hbox{ \leavevmode
\hbox to6.39pt{\vbox to7.98pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower
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\pgfsys@invoke{ }\nullfont\hbox to0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.83333
pt}\pgfsys@lineto{0.0pt}{7.14444pt}\pgfsys@lineto{6.38892pt}{7.14444pt}
\pgfsys@lineto{6.38892pt}{-0.83333pt}\pgfsys@closepath\pgfsys@clipnext
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{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-6.3889pt\hbox{
\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
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0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.83333pt}\pgfsys@lineto{0.0pt}{7.14444pt
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{{}{{}}}{{}{}}{}{{}{}}
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\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
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0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt}
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}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox
{\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
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{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
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{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to
0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.41666pt}\pgfsys@lineto{0.0pt}{3.57222pt
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}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}t_{1}\mathrel{
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###### Theorem 3.1 .

#### 3.3 Estimating the Compounding Decoding Error Using Girsanov’s Theorem

As shown in Eq. (3), computing $\mathcal{E}_{\mathrm{CDE}}(s\shortrightarrow t|\mathbf{x}_{s})$ involves determining the KL divergence between the true distribution and the approximated distribution from DDM’s parallel sampler, which is often intractable. To address this, we treat the ground truth reverse process and the sampling process from DDM’s parallel samplers (introduced in Section 2.2) as two CTMCs, starting from the same initial distribution. By applying Girsanov’s theorem (Ding & Ning, ; Chen et al., ), we derive a tractable formula to compare the KL divergence between the distributions of these stochastic processes at any time interval $[s,t]$. We summarize this as the following general theorem applicable to any two backward CTMCs with $R^{1}_{t}$ and $R^{2}_{t}$ as their respective transition rate matrices:

$$ $\begin{cases}\textrm{CTMC 1}:&q^{1}_{u-du|u}(y\mid x)=\delta_{xy}+R^{1}_{t}(x, y)du+o(du),\\ \textrm{CTMC 2}:&q^{2}_{u-du|u}(y\mid x)=\delta_{xy}+R^{2}_{t}(x,y)du+o(du). \end{cases}$ $$

We defer its proof to Appendix A.2.

###### Theorem 3.2 .

(KL-Divergence Upper Bound, KLUB)
Consider an interval $[s,t]$ ($s>t$). If both CTMCs start from the same initial distribution, $\pi_{s}=\mathbb{P}_{s}=\mathbb{Q}_{s}$, then we have:

$$ $\mathcal{D}_{\mathrm{KL}}(\mathbb{P}_{t}\|\mathbb{Q}_{t})\leq\mathcal{D}_{ \mathrm{KL}}(\mathbb{P}_{\mathrm{paths}}\|\mathbb{Q}_{\mathrm{paths}})= \underbrace{\mathbb{E}_{\mathbb{P}_{\mathrm{paths}}}\left[\sum_{i\neq j}\sum_{ \begin{subarray}{c}t<u\leq s\end{subarray}}H_{u}^{ij}\log\frac{R^{1}_{u}(i,j)} {R^{2}_{u}(i,j)}\right]}_{\triangleq~{}\textrm{{KLUB}}_{\pi_{s}}(\mathbb{P}_{t }\|\mathbb{Q}_{t})}.$ (8) $$

Here, $\mathbb{P}_{t}$ and $\mathbb{Q}_{t}$ are the probability distributions at time $t$ resulting from CTMC 1 and CTMC 2, respectively. $\mathbb{P}_{\mathrm{paths}}$ and $\mathbb{Q}_{\mathrm{paths}}$ denote the distributions over their path spaces $(X_{u})_{u\in[t,s]}$, generated by CTMC 1 and CTMC 2, respectively. The indicator function $H_{u}^{ij}$ is defined as $H_{u}^{ij}=1$ if a transition from state $i$ to $j$ occurs at time $u$, and $H_{u}^{ij}=0$ otherwise.

We denote the rightmost term in Eq. (8) as the *Kullback-Leibler Divergence Upper Bound (KLUB)*, which quantifies the mismatch between distributions generated by different CTMCs based on their rate matrices. Theorem 3.2 demonstrates that the difference in trajectories sampled from different CTMCs depends on the ratio of their rate matrices.

Consider CTMC 1 as the ground truth reverse CTMC and CTMC 2 as the reverse CTMC obtained via parallel sampling by substituting the forward transition kernel with the corresponding reverse-in-time kernel. Thus, $\mathcal{E}_{\mathrm{CDE}}(s\shortrightarrow t|\mathbf{x}_{s})$ can be expressed as the KL divergence between the output distributions of the two CTMCs at time $t$, starting from the initial point $\mathbf{x}_{s}$. This leads to the upper bound (Proof in Appendix A.3):

$$ $\mathcal{E}_{\mathrm{CDE}}(s\shortrightarrow t)\leq\textrm{{KLUB}}_{\mathbb{P} _{s}}(\mathbb{P}_{t}\|\mathbb{Q}^{s\mathrel{\mathchoice{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.13pt{\vbox to3.91pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.40833pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.40833pt}\pgfsys@lineto{0.0pt}{3.50076pt }\pgfsys@lineto{3.13055pt}{3.50076pt}\pgfsys@lineto{3.13055pt}{-0.40833pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.13055pt\hbox {\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to2.24pt{\vbox to2.79pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt }\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox {\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}t}_{t}),$ (9) $$

where $\mathbb{Q}^{s\mathrel{\mathchoice{\mkern 2.0mu\leavevmode\leavevmode\hbox{
\leavevmode\hbox to4.47pt{\vbox to5.58pt{\pgfpicture\makeatletter\hbox{\hskip 0
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{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt}\pgfsys@closepath
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0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox
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{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
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0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.13055pt\hbox
{\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
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This expectation considers the distribution of states at time $s$ and offers a more comprehensive measure of the CDE over the interval $[s,t]$.
Moreover, from the additivity of KLUB, we can bound the sum of CDEs as

$$ $\mathcal{E}_{\mathrm{CDE}}(s\shortrightarrow t)+\mathcal{E}_{\mathrm{CDE}}(t \shortrightarrow u)\leq\textrm{{KLUB}}_{\mathbb{P}_{s}}(\mathbb{P}_{t}\| \mathbb{Q}^{s\mathrel{\mathchoice{\mkern 2.0mu\leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{\pgfpicture\makeatletter\hbox{\hskip 0 .0pt\lower-0.58333pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0} \pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ } \pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{{}{}{}{} \pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt}\pgfsys@lineto {4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt}\pgfsys@closepath \pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}}{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.13pt{\vbox to3.91pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.40833pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.40833pt}\pgfsys@lineto{0.0pt}{3.50076pt }\pgfsys@lineto{3.13055pt}{3.50076pt}\pgfsys@lineto{3.13055pt}{-0.40833pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} 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\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox {\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}t}_{t})+\textrm{{ KLUB}}_{\mathbb{P}_{t}}(\mathbb{P}_{u}\|\mathbb{Q}^{t\mathrel{\mathchoice{ \mkern 2.0mu\leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to 5.58pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt {\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.13pt{\vbox to3.91pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.40833pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ 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}\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt }\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox {\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}u}_{u})=\textrm{{ KLUB}}_{\mathbb{P}_{s}}(\mathbb{P}_{u}\|\mathbb{Q}^{s\mathrel{\mathchoice{ \mkern 2.0mu\leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to 5.58pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt {\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.13pt{\vbox to3.91pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.40833pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.40833pt}\pgfsys@lineto{0.0pt}{3.50076pt }\pgfsys@lineto{3.13055pt}{3.50076pt}\pgfsys@lineto{3.13055pt}{-0.40833pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.13055pt\hbox {\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to2.24pt{\vbox to2.79pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt }\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox {\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}t\mathrel{ \mathchoice{\mkern 2.0mu\leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt {\vbox to5.58pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt \hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{ rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ } \pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt} \pgfsys@invoke{ }\nullfont\hbox to0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333 pt}\pgfsys@lineto{0.0pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{-0.58333pt}\pgfsys@closepath\pgfsys@clipnext \pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}}{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.13pt{\vbox to3.91pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.40833pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.40833pt}\pgfsys@lineto{0.0pt}{3.50076pt }\pgfsys@lineto{3.13055pt}{3.50076pt}\pgfsys@lineto{3.13055pt}{-0.40833pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.13055pt\hbox {\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to2.24pt{\vbox to2.79pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt }\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox {\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}u}_{u}),$ (10) $$

which shows that the KLUB can be useful for comparing the quality of
discretization of the interval $[s,u]$ with different break point $t$.
We can easily extend this result for the sum of CDEs over the entire sampling schedule $\{T=t_{0}\mathrel{\mathchoice{\mkern 2.0mu\leavevmode\leavevmode\hbox{
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\pgfsys@moveto{0.0pt}{-0.83333pt}\pgfsys@lineto{0.0pt}{7.14444pt}
\pgfsys@lineto{6.38892pt}{7.14444pt}\pgfsys@lineto{6.38892pt}{-0.83333pt}
\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-6.3889pt\hbox{
\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
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\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to
0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.83333pt}\pgfsys@lineto{0.0pt}{7.14444pt
}\pgfsys@lineto{6.38892pt}{7.14444pt}\pgfsys@lineto{6.38892pt}{-0.83333pt}
\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-6.3889pt\hbox{
\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
\leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{
\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{
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\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to
0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt}
\pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt}
\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox
{\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
\leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.19pt{\vbox to3.99pt{
\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.41666pt\hbox to0.0pt{
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0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{ }\pgfsys@color@gray@fill{0}
\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to
0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.41666pt}\pgfsys@lineto{0.0pt}{3.57222pt
}\pgfsys@lineto{3.19446pt}{3.57222pt}\pgfsys@lineto{3.19446pt}{-0.41666pt}
\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.19444pt\hbox
{\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}t_{1}\mathrel{
\mathchoice{\mkern 2.0mu\leavevmode\leavevmode\hbox{ \leavevmode\hbox to6.39pt
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\pgfsys@lineto{6.38892pt}{-0.83333pt}\pgfsys@closepath\pgfsys@clipnext
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{{}{{}}}{{}{}}{}{{}{}}
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0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-6.3889pt\hbox{
\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
\leavevmode\leavevmode\hbox{ \leavevmode\hbox to6.39pt{\vbox to7.98pt{
\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.83333pt\hbox to0.0pt{
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0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.83333pt}\pgfsys@lineto{0.0pt}{7.14444pt
}\pgfsys@lineto{6.38892pt}{7.14444pt}\pgfsys@lineto{6.38892pt}{-0.83333pt}
\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-6.3889pt\hbox{
\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
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{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
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{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
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$$ $\mathcal{D}_{\mathrm{KL}}(\mathbb{P}_{0}\|\mathbb{Q}_{0}^{T\mathrel{ \mathchoice{\mkern 2.0mu\leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt {\vbox to5.58pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt \hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{ pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{ } \pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt} \pgfsys@invoke{ }\nullfont\hbox to0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333 pt}\pgfsys@lineto{0.0pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{-0.58333pt}\pgfsys@closepath\pgfsys@clipnext \pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}}{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope 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\pgfsys@lineto{4.47223pt}{-0.58333pt}\pgfsys@closepath\pgfsys@clipnext \pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}}{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{ 0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{ }\pgfsys@color@gray@fill{0} \pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.13pt{\vbox to3.91pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.40833pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{ 0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{ }\pgfsys@color@gray@fill{0} \pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.40833pt}\pgfsys@lineto{0.0pt}{3.50076pt }\pgfsys@lineto{3.13055pt}{3.50076pt}\pgfsys@lineto{3.13055pt}{-0.40833pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.13055pt\hbox {\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to2.24pt{\vbox to2.79pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{ 0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{ }\pgfsys@color@gray@fill{0} \pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt }\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox {\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}\cdots\mathrel{ \mathchoice{\mkern 2.0mu\leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt {\vbox to5.58pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt \hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{ pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{ } \pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt} \pgfsys@invoke{ }\nullfont\hbox to0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333 pt}\pgfsys@lineto{0.0pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{-0.58333pt}\pgfsys@closepath\pgfsys@clipnext \pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}}{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{ 0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{ }\pgfsys@color@gray@fill{0} \pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.13pt{\vbox to3.91pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.40833pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{ 0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{ }\pgfsys@color@gray@fill{0} \pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.40833pt}\pgfsys@lineto{0.0pt}{3.50076pt }\pgfsys@lineto{3.13055pt}{3.50076pt}\pgfsys@lineto{3.13055pt}{-0.40833pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.13055pt\hbox {\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to2.24pt{\vbox to2.79pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{ 0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{ }\pgfsys@color@gray@fill{0} \pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt }\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox {\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}0})\leq\sum_{i=0}^{N -1}\mathcal{E}_{\mathrm{CDE}}(t_{i}\shortrightarrow t_{i+1})\leq\textrm{{KLUB} }_{\mathbb{P}_{T}}(\mathbb{P}_{0}\|\mathbb{Q}^{T\mathrel{\mathchoice{\mkern 2. 0mu\leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{ 0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{ }\pgfsys@color@gray@fill{0} \pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{ 0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{ }\pgfsys@color@gray@fill{0} \pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.13pt{\vbox to3.91pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.40833pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{ 0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{ }\pgfsys@color@gray@fill{0} \pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.40833pt}\pgfsys@lineto{0.0pt}{3.50076pt }\pgfsys@lineto{3.13055pt}{3.50076pt}\pgfsys@lineto{3.13055pt}{-0.40833pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.13055pt\hbox {\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to2.24pt{\vbox to2.79pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{ 0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{ }\pgfsys@color@gray@fill{0} \pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt }\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox {\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}t_{1}\mathrel{ \mathchoice{\mkern 2.0mu\leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt {\vbox to5.58pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt \hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{ pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{ } \pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt} \pgfsys@invoke{ }\nullfont\hbox to0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333 pt}\pgfsys@lineto{0.0pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{-0.58333pt}\pgfsys@closepath\pgfsys@clipnext \pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}}{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{ 0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{ }\pgfsys@color@gray@fill{0} \pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.13pt{\vbox to3.91pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.40833pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{ 0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{ }\pgfsys@color@gray@fill{0} \pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.40833pt}\pgfsys@lineto{0.0pt}{3.50076pt }\pgfsys@lineto{3.13055pt}{3.50076pt}\pgfsys@lineto{3.13055pt}{-0.40833pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.13055pt\hbox {\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to2.24pt{\vbox to2.79pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{ 0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{ }\pgfsys@color@gray@fill{0} \pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt }\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox {\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}\cdots\mathrel{ \mathchoice{\mkern 2.0mu\leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt {\vbox to5.58pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt \hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{ pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{ } \pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt} \pgfsys@invoke{ }\nullfont\hbox to0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333 pt}\pgfsys@lineto{0.0pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{-0.58333pt}\pgfsys@closepath\pgfsys@clipnext \pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}}{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{ 0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{ }\pgfsys@color@gray@fill{0} \pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.13pt{\vbox to3.91pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.40833pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{ 0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{ }\pgfsys@color@gray@fill{0} \pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.40833pt}\pgfsys@lineto{0.0pt}{3.50076pt }\pgfsys@lineto{3.13055pt}{3.50076pt}\pgfsys@lineto{3.13055pt}{-0.40833pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.13055pt\hbox {\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to2.24pt{\vbox to2.79pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{ 0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{ }\pgfsys@color@gray@fill{0} \pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt }\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox {\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}0}_{0}),$ (11) $$

where inequality on the left-hand side comes from Theorem 3.1.

To summarize, optimizing the sampling schedule involves finding a set of timesteps $\{T\mathrel{\mathchoice{\mkern 2.0mu\leavevmode\leavevmode\hbox{ \leavevmode
\hbox to6.39pt{\vbox to7.98pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower
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\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}
\pgfsys@invoke{ }\nullfont\hbox to0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.83333
pt}\pgfsys@lineto{0.0pt}{7.14444pt}\pgfsys@lineto{6.38892pt}{7.14444pt}
\pgfsys@lineto{6.38892pt}{-0.83333pt}\pgfsys@closepath\pgfsys@clipnext
\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-6.3889pt\hbox{
\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
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\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.83333pt\hbox to0.0pt{
\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{
0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{ }\pgfsys@color@gray@fill{0}
\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to
0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.83333pt}\pgfsys@lineto{0.0pt}{7.14444pt
}\pgfsys@lineto{6.38892pt}{7.14444pt}\pgfsys@lineto{6.38892pt}{-0.83333pt}
\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-6.3889pt\hbox{
\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
\leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{
\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{
\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{
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\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to
0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt}
\pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt}
\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox
{\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
\leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.19pt{\vbox to3.99pt{
\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.41666pt\hbox to0.0pt{
\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{
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\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to
0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.41666pt}\pgfsys@lineto{0.0pt}{3.57222pt
}\pgfsys@lineto{3.19446pt}{3.57222pt}\pgfsys@lineto{3.19446pt}{-0.41666pt}
\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.19444pt\hbox
{\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}t_{1}\mathrel{
\mathchoice{\mkern 2.0mu\leavevmode\leavevmode\hbox{ \leavevmode\hbox to6.39pt
{\vbox to7.98pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.83333pt
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{{}{{}}}{{}{}}{}{{}{}}
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\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
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\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-6.3889pt\hbox{
\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
\leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{
\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{
\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{
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\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to
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{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
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{\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
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\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
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\mathchoice{\mkern 2.0mu\leavevmode\leavevmode\hbox{ \leavevmode\hbox to6.39pt
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{{}{{}}}{{}{}}{}{{}{}}
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{{}{{}}}{{}{}}{}{{}{}}
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{\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}t_{N-1}\mathrel{
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\pgfsys@lineto{6.38892pt}{-0.83333pt}\pgfsys@closepath\pgfsys@clipnext
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{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
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\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
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{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
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\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-6.3889pt\hbox{
\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
\leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{
\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{
\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{
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\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to
0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt}
\pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt}
\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox
{\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
\leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.19pt{\vbox to3.99pt{
\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.41666pt\hbox to0.0pt{
\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{
0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{ }\pgfsys@color@gray@fill{0}
\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to
0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.41666pt}\pgfsys@lineto{0.0pt}{3.57222pt
}\pgfsys@lineto{3.19446pt}{3.57222pt}\pgfsys@lineto{3.19446pt}{-0.41666pt}
\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.19444pt\hbox
{\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}0\}$ that minimizes the KLUB on the right-hand side. This approach approximately reduces the cumulative CDE (middle) and provides an upper bound on the KL divergence between the true distribution and the sampled distribution for the given schedule (left-hand side).

###### Theorem 3.2 .

#### 3.4 Feasible Computation with Hierarchical Breakdown Strategy

Using the derived KLUB, we can formulate the timestep search as a minimization problem over KLUB. Here, we employ a hierarchical breakdown strategy, dividing a coarser sampling schedule into a finer one, as shown in Figure 3. Suppose our sampling schedule is given by $\{T\mathrel{\mathchoice{\mkern 2.0mu\leavevmode\leavevmode\hbox{ \leavevmode
\hbox to6.39pt{\vbox to7.98pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower
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\pgfsys@invoke{ }\nullfont\hbox to0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.83333
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\pgfsys@lineto{6.38892pt}{-0.83333pt}\pgfsys@closepath\pgfsys@clipnext
\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-6.3889pt\hbox{
\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
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{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
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{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
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\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
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{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
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{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
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\pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt}
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{{}{{}}}{{}{}}{}{{}{}}
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{\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
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{{}{{}}}{{}{}}{}{{}{}}
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{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
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\mathchoice{\mkern 2.0mu\leavevmode\leavevmode\hbox{ \leavevmode\hbox to6.39pt
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\pgfsys@lineto{6.38892pt}{-0.83333pt}\pgfsys@closepath\pgfsys@clipnext
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{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
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{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
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{{}{{}}}{{}{}}{}{{}{}}
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{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
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\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
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{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}0\}$. Let $\mathbb{Q}_{0}^{T\mathrel{\mathchoice{\mkern 2.0mu\leavevmode\leavevmode\hbox{
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\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt}\pgfsys@lineto
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\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox
{\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
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{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
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\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}
{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
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\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
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{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
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{{}{{}}}{{}{}}{}{{}{}}
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to
0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt
}\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt}
\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox
{\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}t\mathrel{
\mathchoice{\mkern 2.0mu\leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt
{\vbox to5.58pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt
\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{
rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }
\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}
\pgfsys@invoke{ }\nullfont\hbox to0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333
pt}\pgfsys@lineto{0.0pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{5.0011pt}
\pgfsys@lineto{4.47223pt}{-0.58333pt}\pgfsys@closepath\pgfsys@clipnext
\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox
{\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
\leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{
\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{
\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}
{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to
0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt}
\pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt}
\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox
{\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
\leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.13pt{\vbox to3.91pt{
\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.40833pt\hbox to0.0pt{
\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}
{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to
0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.40833pt}\pgfsys@lineto{0.0pt}{3.50076pt
}\pgfsys@lineto{3.13055pt}{3.50076pt}\pgfsys@lineto{3.13055pt}{-0.40833pt}
\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.13055pt\hbox
{\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
\leavevmode\leavevmode\hbox{ \leavevmode\hbox to2.24pt{\vbox to2.79pt{
\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{
\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}
{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to
0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt
}\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt}
\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox
{\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}0}$ represent the distribution generated by this schedule. Our goal is to find the optimal $t$ that minimizes cumulative CDE, i.e.,
$\mathcal{E}_{\mathrm{CDE}}(T\shortrightarrow t)+\mathcal{E}_{\mathrm{CDE}}(t
\shortrightarrow 0)$.
This is approximately achievable by minimizing its KLUB upper bound:

$$ $t_{1}=\operatorname*{arg\,min}_{t\in(T,0)}\textrm{{KLUB}}(\mathbb{P}_{0}\| \mathbb{Q}_{0}^{T\mathrel{\mathchoice{\mkern 2.0mu\leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{\pgfpicture\makeatletter\hbox{\hskip 0 .0pt\lower-0.58333pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0} \pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ } \pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{{}{}{}{} \pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt}\pgfsys@lineto {4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt}\pgfsys@closepath \pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}}{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.13pt{\vbox to3.91pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.40833pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.40833pt}\pgfsys@lineto{0.0pt}{3.50076pt }\pgfsys@lineto{3.13055pt}{3.50076pt}\pgfsys@lineto{3.13055pt}{-0.40833pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.13055pt\hbox {\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to2.24pt{\vbox to2.79pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt }\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox {\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}t\mathrel{ \mathchoice{\mkern 2.0mu\leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt {\vbox to5.58pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt \hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{ rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ } \pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt} \pgfsys@invoke{ }\nullfont\hbox to0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333 pt}\pgfsys@lineto{0.0pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{-0.58333pt}\pgfsys@closepath\pgfsys@clipnext \pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}}{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.13pt{\vbox to3.91pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.40833pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.40833pt}\pgfsys@lineto{0.0pt}{3.50076pt }\pgfsys@lineto{3.13055pt}{3.50076pt}\pgfsys@lineto{3.13055pt}{-0.40833pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.13055pt\hbox {\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to2.24pt{\vbox to2.79pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt }\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox {\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}0})$ (12) $$

With the initial refined interval $\{T\mathrel{\mathchoice{\mkern 2.0mu\leavevmode\leavevmode\hbox{ \leavevmode
\hbox to6.39pt{\vbox to7.98pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower
-0.83333pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{
pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }
\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}
\pgfsys@invoke{ }\nullfont\hbox to0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.83333
pt}\pgfsys@lineto{0.0pt}{7.14444pt}\pgfsys@lineto{6.38892pt}{7.14444pt}
\pgfsys@lineto{6.38892pt}{-0.83333pt}\pgfsys@closepath\pgfsys@clipnext
\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-6.3889pt\hbox{
\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
\leavevmode\leavevmode\hbox{ \leavevmode\hbox to6.39pt{\vbox to7.98pt{
\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.83333pt\hbox to0.0pt{
\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}
{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to
0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.83333pt}\pgfsys@lineto{0.0pt}{7.14444pt
}\pgfsys@lineto{6.38892pt}{7.14444pt}\pgfsys@lineto{6.38892pt}{-0.83333pt}
\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-6.3889pt\hbox{
\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
\leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{
\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{
\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}
{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to
0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt}
\pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt}
\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox
{\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
\leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.19pt{\vbox to3.99pt{
\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.41666pt\hbox to0.0pt{
\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}
{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to
0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.41666pt}\pgfsys@lineto{0.0pt}{3.57222pt
}\pgfsys@lineto{3.19446pt}{3.57222pt}\pgfsys@lineto{3.19446pt}{-0.41666pt}
\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.19444pt\hbox
{\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}t_{t}\mathrel{
\mathchoice{\mkern 2.0mu\leavevmode\leavevmode\hbox{ \leavevmode\hbox to6.39pt
{\vbox to7.98pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.83333pt
\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{
rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }
\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}
\pgfsys@invoke{ }\nullfont\hbox to0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.83333
pt}\pgfsys@lineto{0.0pt}{7.14444pt}\pgfsys@lineto{6.38892pt}{7.14444pt}
\pgfsys@lineto{6.38892pt}{-0.83333pt}\pgfsys@closepath\pgfsys@clipnext
\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-6.3889pt\hbox{
\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
\leavevmode\leavevmode\hbox{ \leavevmode\hbox to6.39pt{\vbox to7.98pt{
\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.83333pt\hbox to0.0pt{
\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}
{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to
0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.83333pt}\pgfsys@lineto{0.0pt}{7.14444pt
}\pgfsys@lineto{6.38892pt}{7.14444pt}\pgfsys@lineto{6.38892pt}{-0.83333pt}
\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-6.3889pt\hbox{
\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
\leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{
\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{
\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}
{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to
0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt}
\pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt}
\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox
{\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
\leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.19pt{\vbox to3.99pt{
\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.41666pt\hbox to0.0pt{
\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}
{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to
0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.41666pt}\pgfsys@lineto{0.0pt}{3.57222pt
}\pgfsys@lineto{3.19446pt}{3.57222pt}\pgfsys@lineto{3.19446pt}{-0.41666pt}
\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.19444pt\hbox
{\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}0\}$, we seek optimal timesteps $t_{2}\in(T,t_{1})$ and $t_{3}\in(t_{1},0)$ by solving the following minimization problems:

$$ $t_{2}\in\operatorname*{arg\,min}_{t\in(T,t_{1})}\textrm{{KLUB}}(\mathbb{P}_{t_ {1}}\|\mathbb{Q}_{t_{1}}^{T\mathrel{\mathchoice{\mkern 2.0mu\leavevmode \leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{\pgfpicture \makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.13pt{\vbox to3.91pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.40833pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.40833pt}\pgfsys@lineto{0.0pt}{3.50076pt }\pgfsys@lineto{3.13055pt}{3.50076pt}\pgfsys@lineto{3.13055pt}{-0.40833pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.13055pt\hbox {\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to2.24pt{\vbox to2.79pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt }\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox {\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}t\mathrel{ \mathchoice{\mkern 2.0mu\leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt {\vbox to5.58pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt \hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{ rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ } \pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt} \pgfsys@invoke{ }\nullfont\hbox to0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333 pt}\pgfsys@lineto{0.0pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{-0.58333pt}\pgfsys@closepath\pgfsys@clipnext \pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}}{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.13pt{\vbox to3.91pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.40833pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.40833pt}\pgfsys@lineto{0.0pt}{3.50076pt }\pgfsys@lineto{3.13055pt}{3.50076pt}\pgfsys@lineto{3.13055pt}{-0.40833pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.13055pt\hbox {\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to2.24pt{\vbox to2.79pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt }\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox {\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}t_{1}})\quad\text{ and}\quad t_{3}\in\operatorname*{arg\,min}_{t\in(t_{1},0)}\textrm{{KLUB}}( \mathbb{P}_{0}\|\mathbb{Q}_{0}^{t_{1}\mathrel{\mathchoice{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} 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Figure: Figure 3: We optimize the sampling schedule by refining it from coarse intervals to finer intervals, using a hierarchical breakdown strategy.
Refer to caption: x2.png

The first minimization problem targets matching $\mathcal{D}_{\mathrm{KL}}(\mathbb{P}_{t_{1}}\|\mathbb{Q}_{t_{1}})$, while the second focuses on matching $\mathcal{D}_{\mathrm{KL}}(\mathbb{P}_{0}\|\mathbb{Q}_{0})$.
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{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}t_{3}\mathrel{
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{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
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{\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
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{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
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{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
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0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.41666pt}\pgfsys@lineto{0.0pt}{3.57222pt
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\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.19444pt\hbox
{\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}0\}$. By iterating this process, we continue splitting each interval into smaller ones, optimizing breakpoints using the KLUB criterion. After $K$ iterations, this hierarchical strategy yields a sampling schedule with $2^{K}$ NFEs, optimizing the schedule as the number of steps increases.

#### 3.5 Feasible Computation for KLUB Estimation

Directly estimating $\textrm{{KLUB}}_{\mathbb{P}_{T}}(\mathbb{P}_{0}\|\mathbb{Q}_{0}^{T\mathrel{
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{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
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{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
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{\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
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{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
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{{}{{}}}{{}{}}{}{{}{}}
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}t\mathrel{
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{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
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{\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
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{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
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{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
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{\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
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\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}
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{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
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\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox
{\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}0})$ by using Eq. (8) is practically infeasible due to computational complexities. To address this issue, we propose two techniques that simplify the estimation process.

##### Technique 1: Maximizing KL Divergence from a Coarser Approximation

Instead of minimizing the mismatch between the ground truth distribution $\mathbb{P}_{\text{paths}}$ and $\mathbb{Q}_{\text{paths}}^{T\mathrel{\mathchoice{\mkern 2.0mu\leavevmode
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{{}{{}}}{{}{}}{}{{}{}}
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{\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
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}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox
{\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
\leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.13pt{\vbox to3.91pt{
\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.40833pt\hbox to0.0pt{
\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}
{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to
0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.40833pt}\pgfsys@lineto{0.0pt}{3.50076pt
}\pgfsys@lineto{3.13055pt}{3.50076pt}\pgfsys@lineto{3.13055pt}{-0.40833pt}
\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.13055pt\hbox
{\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
\leavevmode\leavevmode\hbox{ \leavevmode\hbox to2.24pt{\vbox to2.79pt{
\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{
\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}
{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to
0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt
}\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt}
\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox
{\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}t\mathrel{
\mathchoice{\mkern 2.0mu\leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt
{\vbox to5.58pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt
\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{
rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }
\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}
\pgfsys@invoke{ }\nullfont\hbox to0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333
pt}\pgfsys@lineto{0.0pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{5.0011pt}
\pgfsys@lineto{4.47223pt}{-0.58333pt}\pgfsys@closepath\pgfsys@clipnext
\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox
{\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
\leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{
\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{
\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}
{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to
0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt}
\pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt}
\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox
{\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
\leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.13pt{\vbox to3.91pt{
\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.40833pt\hbox to0.0pt{
\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}
{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to
0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.40833pt}\pgfsys@lineto{0.0pt}{3.50076pt
}\pgfsys@lineto{3.13055pt}{3.50076pt}\pgfsys@lineto{3.13055pt}{-0.40833pt}
\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.13055pt\hbox
{\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
\leavevmode\leavevmode\hbox{ \leavevmode\hbox to2.24pt{\vbox to2.79pt{
\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{
\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}
{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to
0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt
}\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt}
\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox
{\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}0}$, we choose to maximize the discrepancy between $\mathbb{Q}_{\text{paths}}^{T\mathrel{\mathchoice{\mkern 2.0mu\leavevmode
\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{\pgfpicture
\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{
\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}
{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to
0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt}
\pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt}
\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox
{\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
\leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{
\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{
\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}
{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to
0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt}
\pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt}
\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox
{\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
\leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.13pt{\vbox to3.91pt{
\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.40833pt\hbox to0.0pt{
\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}
{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to
0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.40833pt}\pgfsys@lineto{0.0pt}{3.50076pt
}\pgfsys@lineto{3.13055pt}{3.50076pt}\pgfsys@lineto{3.13055pt}{-0.40833pt}
\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.13055pt\hbox
{\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
\leavevmode\leavevmode\hbox{ \leavevmode\hbox to2.24pt{\vbox to2.79pt{
\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{
\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}
{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to
0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt
}\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt}
\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox
{\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}t\mathrel{
\mathchoice{\mkern 2.0mu\leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt
{\vbox to5.58pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt
\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{
rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }
\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}
\pgfsys@invoke{ }\nullfont\hbox to0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333
pt}\pgfsys@lineto{0.0pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{5.0011pt}
\pgfsys@lineto{4.47223pt}{-0.58333pt}\pgfsys@closepath\pgfsys@clipnext
\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox
{\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
\leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{
\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{
\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}
{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to
0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt}
\pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt}
\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox
{\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
\leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.13pt{\vbox to3.91pt{
\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.40833pt\hbox to0.0pt{
\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}
{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to
0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.40833pt}\pgfsys@lineto{0.0pt}{3.50076pt
}\pgfsys@lineto{3.13055pt}{3.50076pt}\pgfsys@lineto{3.13055pt}{-0.40833pt}
\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.13055pt\hbox
{\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
\leavevmode\leavevmode\hbox{ \leavevmode\hbox to2.24pt{\vbox to2.79pt{
\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{
\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}
{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to
0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt
}\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt}
\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox
{\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}0}$ and a simpler, coarser approximation $\mathbb{Q}_{\text{paths}}^{T\mathrel{\mathchoice{\mkern 2.0mu\leavevmode
\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{\pgfpicture
\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{
\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}
{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to
0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt}
\pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt}
\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox
{\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
\leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{
\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{
\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}
{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to
0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt}
\pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt}
\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox
{\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
\leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.13pt{\vbox to3.91pt{
\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.40833pt\hbox to0.0pt{
\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}
{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to
0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.40833pt}\pgfsys@lineto{0.0pt}{3.50076pt
}\pgfsys@lineto{3.13055pt}{3.50076pt}\pgfsys@lineto{3.13055pt}{-0.40833pt}
\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.13055pt\hbox
{\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
\leavevmode\leavevmode\hbox{ \leavevmode\hbox to2.24pt{\vbox to2.79pt{
\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{
\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}
{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to
0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt
}\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt}
\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox
{\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}0}$. The key insight is that maximizing this divergence, we can find the optimal sampling time $t$, which helps reduce the compounding error relative to the true distribution. This relationship can be approximated as (Appendix A.4):

$$ $\mathcal{D}_{\mathrm{KL}}(\mathbb{Q}_{\text{paths}}^{T\mathrel{\mathchoice{ \mkern 2.0mu\leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to 5.58pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt {\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.13pt{\vbox to3.91pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.40833pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.40833pt}\pgfsys@lineto{0.0pt}{3.50076pt }\pgfsys@lineto{3.13055pt}{3.50076pt}\pgfsys@lineto{3.13055pt}{-0.40833pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.13055pt\hbox {\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to2.24pt{\vbox to2.79pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt }\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox {\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}t\mathrel{ \mathchoice{\mkern 2.0mu\leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt {\vbox to5.58pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt \hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{ rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ } \pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt} \pgfsys@invoke{ }\nullfont\hbox to0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333 pt}\pgfsys@lineto{0.0pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{-0.58333pt}\pgfsys@closepath\pgfsys@clipnext \pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}}{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.13pt{\vbox to3.91pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.40833pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.40833pt}\pgfsys@lineto{0.0pt}{3.50076pt }\pgfsys@lineto{3.13055pt}{3.50076pt}\pgfsys@lineto{3.13055pt}{-0.40833pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.13055pt\hbox {\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to2.24pt{\vbox to2.79pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt }\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox {\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}0}\|\mathbb{Q}_{ \text{paths}}^{T\mathrel{\mathchoice{\mkern 2.0mu\leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{\pgfpicture\makeatletter\hbox{\hskip 0 .0pt\lower-0.58333pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0} \pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ } \pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{{}{}{}{} \pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt}\pgfsys@lineto {4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt}\pgfsys@closepath \pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}}{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.13pt{\vbox to3.91pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.40833pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.40833pt}\pgfsys@lineto{0.0pt}{3.50076pt }\pgfsys@lineto{3.13055pt}{3.50076pt}\pgfsys@lineto{3.13055pt}{-0.40833pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.13055pt\hbox {\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to2.24pt{\vbox to2.79pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt }\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox {\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}0})\approx\mathcal{D }_{\mathrm{KL}}(\mathbb{P}_{\text{paths}}\|\mathbb{Q}_{\text{paths}}^{T \mathrel{\mathchoice{\mkern 2.0mu\leavevmode\leavevmode\hbox{ \leavevmode\hbox to 4.47pt{\vbox to5.58pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333 pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor }{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ } \pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt} \pgfsys@invoke{ }\nullfont\hbox to0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333 pt}\pgfsys@lineto{0.0pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{-0.58333pt}\pgfsys@closepath\pgfsys@clipnext \pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}}{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.13pt{\vbox to3.91pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.40833pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.40833pt}\pgfsys@lineto{0.0pt}{3.50076pt }\pgfsys@lineto{3.13055pt}{3.50076pt}\pgfsys@lineto{3.13055pt}{-0.40833pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.13055pt\hbox {\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to2.24pt{\vbox to2.79pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt }\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox {\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}0})-\mathcal{D}_{ \mathrm{KL}}(\mathbb{P}_{\text{paths}}\|\mathbb{Q}_{\text{paths}}^{T\mathrel{ \mathchoice{\mkern 2.0mu\leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt {\vbox to5.58pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt \hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{ rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ } \pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt} \pgfsys@invoke{ }\nullfont\hbox to0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333 pt}\pgfsys@lineto{0.0pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{-0.58333pt}\pgfsys@closepath\pgfsys@clipnext \pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}}{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.13pt{\vbox to3.91pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.40833pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.40833pt}\pgfsys@lineto{0.0pt}{3.50076pt }\pgfsys@lineto{3.13055pt}{3.50076pt}\pgfsys@lineto{3.13055pt}{-0.40833pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.13055pt\hbox {\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to2.24pt{\vbox to2.79pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt }\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox {\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}t\mathrel{ \mathchoice{\mkern 2.0mu\leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt {\vbox to5.58pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt \hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{ rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ } \pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt} \pgfsys@invoke{ }\nullfont\hbox to0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333 pt}\pgfsys@lineto{0.0pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{-0.58333pt}\pgfsys@closepath\pgfsys@clipnext \pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}}{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.13pt{\vbox to3.91pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.40833pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.40833pt}\pgfsys@lineto{0.0pt}{3.50076pt }\pgfsys@lineto{3.13055pt}{3.50076pt}\pgfsys@lineto{3.13055pt}{-0.40833pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.13055pt\hbox {\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to2.24pt{\vbox to2.79pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt }\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox {\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}0}),$ $$

Thus, by maximizing the divergence on the left-hand side, we effectively minimize the discrepancy $\mathcal{D}_{\mathrm{KL}}(\mathbb{P}_{\text{paths}}\,\big{\|}\,\mathbb{Q}_{
\text{paths}}^{T\mathrel{\mathchoice{\mkern 2.0mu\leavevmode\leavevmode\hbox{
\leavevmode\hbox to4.47pt{\vbox to5.58pt{\pgfpicture\makeatletter\hbox{\hskip 0
.0pt\lower-0.58333pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}
\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }
\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{{}{}{}{}
\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt}\pgfsys@lineto
{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt}\pgfsys@closepath
\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox
{\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
\leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{
\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{
\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}
{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to
0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt}
\pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt}
\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox
{\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
\leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.13pt{\vbox to3.91pt{
\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.40833pt\hbox to0.0pt{
\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}
{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to
0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.40833pt}\pgfsys@lineto{0.0pt}{3.50076pt
}\pgfsys@lineto{3.13055pt}{3.50076pt}\pgfsys@lineto{3.13055pt}{-0.40833pt}
\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.13055pt\hbox
{\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
\leavevmode\leavevmode\hbox{ \leavevmode\hbox to2.24pt{\vbox to2.79pt{
\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{
\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}
{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to
0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt
}\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt}
\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox
{\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}t\mathrel{
\mathchoice{\mkern 2.0mu\leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt
{\vbox to5.58pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt
\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{
rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }
\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}
\pgfsys@invoke{ }\nullfont\hbox to0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333
pt}\pgfsys@lineto{0.0pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{5.0011pt}
\pgfsys@lineto{4.47223pt}{-0.58333pt}\pgfsys@closepath\pgfsys@clipnext
\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
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{\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
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{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
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0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt}
\pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt}
\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
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{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
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{\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
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{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
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{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
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\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
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{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.13055pt\hbox
{\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
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{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
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\leavevmode\leavevmode\hbox{ \leavevmode\hbox to2.24pt{\vbox to2.79pt{
\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{
\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}
{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to
0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt
}\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt}
\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox
{\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}0})$ between the true distribution $\mathbb{P}_{\text{paths}}$ and the optimized schedule $\mathbb{Q}_{\text{paths}}^{T\mathrel{\mathchoice{\mkern 2.0mu\leavevmode
\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{\pgfpicture
\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{
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{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
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0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt}
\pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt}
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{{}{{}}}{{}{}}{}{{}{}}
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{\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
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{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
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{\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
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{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
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{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
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0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.40833pt}\pgfsys@lineto{0.0pt}{3.50076pt
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\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
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{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
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{\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
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{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
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{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
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{\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}t\mathrel{
\mathchoice{\mkern 2.0mu\leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt
{\vbox to5.58pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt
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rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }
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\pgfsys@invoke{ }\nullfont\hbox to0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333
pt}\pgfsys@lineto{0.0pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{5.0011pt}
\pgfsys@lineto{4.47223pt}{-0.58333pt}\pgfsys@closepath\pgfsys@clipnext
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{{}{{}}}{{}{}}{}{{}{}}
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0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox
{\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
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{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
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{\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
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{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
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\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}
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{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to
0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.40833pt}\pgfsys@lineto{0.0pt}{3.50076pt
}\pgfsys@lineto{3.13055pt}{3.50076pt}\pgfsys@lineto{3.13055pt}{-0.40833pt}
\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.13055pt\hbox
{\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
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\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{
\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}
{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to
0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt
}\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt}
\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox
{\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}0}$.

Based on this, we can rewrite our optimization objective as follows (Proof in Appendix A.4):

$$ $t^{*}=\operatorname*{arg\,max}_{t\in(T,0)}\textrm{{KLUB}}_{\mathbb{Q}_{T}}( \mathbb{Q}_{0}^{T\mathrel{\mathchoice{\mkern 2.0mu\leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{\pgfpicture\makeatletter\hbox{\hskip 0 .0pt\lower-0.58333pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0} \pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ } \pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{{}{}{}{} \pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt}\pgfsys@lineto {4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt}\pgfsys@closepath \pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}}{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.13pt{\vbox to3.91pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.40833pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.40833pt}\pgfsys@lineto{0.0pt}{3.50076pt }\pgfsys@lineto{3.13055pt}{3.50076pt}\pgfsys@lineto{3.13055pt}{-0.40833pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} 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\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.20833pt}\pgfsys@lineto{0.0pt}{1.7861pt} \pgfsys@lineto{1.59723pt}{1.7861pt}\pgfsys@lineto{1.59723pt}{-0.20833pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-1.59721pt\hbox {\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}0}}\left[\sum_{i\neq j }\log\frac{R_{t}(i,j)}{R_{T}(i,j)}\sum_{0<u\leq t}H_{u}^{ij}\right]$ (13) $$

Equation Eq. (13) offers a significant computational advantage over Eq. (8) by eliminating the need to compute the rate matrix at every transition time. Note that, calculating the reverse rate matrix involves neural network evaluations. Given a trajectory $(X_{u})_{u\in[T,0]}$, Eq. (8) requires computing the rate matrix $R_{u}$ for all transitions where $H_{u}^{ij}=1$. In contrast, Eq. (13) decouples the rate matrix from the transition times $u$, requiring computations only at times $T$ and $t$. Moreover, between times $T$ and $t$, the rate matrices for both $\mathbb{Q}^{T\mathrel{\mathchoice{\mkern 2.0mu\leavevmode\leavevmode\hbox{
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{\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
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{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
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{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
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{{}{{}}}{{}{}}{}{{}{}}
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\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt}\pgfsys@lineto
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\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
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{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
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\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
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{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}0}_{\text{paths}}$ are equal to $R_{T}$, eliminating the need for additional computations in this interval.

##### Technique 2: Applying the Law of Total Expectation

Instead of directly compute the expectation over $\mathbb{Q}_{\text{paths}}^{T\mathrel{\mathchoice{\mkern 2.0mu\leavevmode
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{{}{{}}}{{}{}}{}{{}{}}
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{\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
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{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
\leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{
\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{
\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}
{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to
0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt}
\pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt}
\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox
{\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
\leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.13pt{\vbox to3.91pt{
\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.40833pt\hbox to0.0pt{
\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}
{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to
0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.40833pt}\pgfsys@lineto{0.0pt}{3.50076pt
}\pgfsys@lineto{3.13055pt}{3.50076pt}\pgfsys@lineto{3.13055pt}{-0.40833pt}
\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.13055pt\hbox
{\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
\leavevmode\leavevmode\hbox{ \leavevmode\hbox to2.24pt{\vbox to2.79pt{
\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{
\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}
{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to
0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt
}\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt}
\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox
{\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}t\mathrel{
\mathchoice{\mkern 2.0mu\leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt
{\vbox to5.58pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt
\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{
rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }
\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}
\pgfsys@invoke{ }\nullfont\hbox to0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333
pt}\pgfsys@lineto{0.0pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{5.0011pt}
\pgfsys@lineto{4.47223pt}{-0.58333pt}\pgfsys@closepath\pgfsys@clipnext
\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox
{\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
\leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{
\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{
\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}
{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to
0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt}
\pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt}
\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox
{\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
\leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.13pt{\vbox to3.91pt{
\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.40833pt\hbox to0.0pt{
\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}
{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to
0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.40833pt}\pgfsys@lineto{0.0pt}{3.50076pt
}\pgfsys@lineto{3.13055pt}{3.50076pt}\pgfsys@lineto{3.13055pt}{-0.40833pt}
\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.13055pt\hbox
{\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
\leavevmode\leavevmode\hbox{ \leavevmode\hbox to2.24pt{\vbox to2.79pt{
\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{
\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}
{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to
0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt
}\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt}
\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox
{\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}0}$, we compute expectation of conditional expectation, using the law of total expectation, i.e., $\mathbb{E}[X]=\mathbb{E}[\mathbb{E}[X|Y]]$:

$$ $\mathbb{E}_{\mathbb{Q}_{\text{paths}}^{T\mathrel{\mathchoice{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.19pt{\vbox to3.99pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.41666pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.41666pt}\pgfsys@lineto{0.0pt}{3.57222pt }\pgfsys@lineto{3.19446pt}{3.57222pt}\pgfsys@lineto{3.19446pt}{-0.41666pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.19444pt\hbox {\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.19pt{\vbox to3.99pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.41666pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.41666pt}\pgfsys@lineto{0.0pt}{3.57222pt }\pgfsys@lineto{3.19446pt}{3.57222pt}\pgfsys@lineto{3.19446pt}{-0.41666pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.19444pt\hbox {\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to2.24pt{\vbox to2.79pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt }\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox {\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to1.6pt{\vbox to1.99pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.20833pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.20833pt}\pgfsys@lineto{0.0pt}{1.7861pt} \pgfsys@lineto{1.59723pt}{1.7861pt}\pgfsys@lineto{1.59723pt}{-0.20833pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-1.59721pt\hbox {\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}t\mathrel{ \mathchoice{\mkern 2.0mu\leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.19pt {\vbox to3.99pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.41666pt \hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{ rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ } \pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt} \pgfsys@invoke{ }\nullfont\hbox to0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.41666 pt}\pgfsys@lineto{0.0pt}{3.57222pt}\pgfsys@lineto{3.19446pt}{3.57222pt} \pgfsys@lineto{3.19446pt}{-0.41666pt}\pgfsys@closepath\pgfsys@clipnext \pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}}{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.19444pt\hbox {\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.19pt{\vbox to3.99pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.41666pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.41666pt}\pgfsys@lineto{0.0pt}{3.57222pt }\pgfsys@lineto{3.19446pt}{3.57222pt}\pgfsys@lineto{3.19446pt}{-0.41666pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.19444pt\hbox {\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to2.24pt{\vbox to2.79pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt }\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox {\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to1.6pt{\vbox to1.99pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.20833pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.20833pt}\pgfsys@lineto{0.0pt}{1.7861pt} \pgfsys@lineto{1.59723pt}{1.7861pt}\pgfsys@lineto{1.59723pt}{-0.20833pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-1.59721pt\hbox {\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}0}}\left[\sum_{i\neq j }\log\frac{R_{t}(i,j)}{R_{T}(i,j)}\sum_{0<u\leq t}H_{u}^{ij}\right]=\mathbb{E} _{\mathbb{Q}_{\text{paths}}^{T\mathrel{\mathchoice{\mkern 2.0mu\leavevmode \leavevmode\hbox{ \leavevmode\hbox to3.19pt{\vbox to3.99pt{\pgfpicture \makeatletter\hbox{\hskip 0.0pt\lower-0.41666pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.41666pt}\pgfsys@lineto{0.0pt}{3.57222pt }\pgfsys@lineto{3.19446pt}{3.57222pt}\pgfsys@lineto{3.19446pt}{-0.41666pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.19444pt\hbox {\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.19pt{\vbox to3.99pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.41666pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.41666pt}\pgfsys@lineto{0.0pt}{3.57222pt }\pgfsys@lineto{3.19446pt}{3.57222pt}\pgfsys@lineto{3.19446pt}{-0.41666pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.19444pt\hbox {\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to2.24pt{\vbox to2.79pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt }\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox {\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to1.6pt{\vbox to1.99pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.20833pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.20833pt}\pgfsys@lineto{0.0pt}{1.7861pt} \pgfsys@lineto{1.59723pt}{1.7861pt}\pgfsys@lineto{1.59723pt}{-0.20833pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-1.59721pt\hbox {\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}t}}\left[\mathbb{E}_ {\mathbb{Q}_{\text{paths}}^{t\mathrel{\mathchoice{\mkern 2.0mu\leavevmode \leavevmode\hbox{ \leavevmode\hbox to3.19pt{\vbox to3.99pt{\pgfpicture \makeatletter\hbox{\hskip 0.0pt\lower-0.41666pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.41666pt}\pgfsys@lineto{0.0pt}{3.57222pt }\pgfsys@lineto{3.19446pt}{3.57222pt}\pgfsys@lineto{3.19446pt}{-0.41666pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.19444pt\hbox {\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.19pt{\vbox to3.99pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.41666pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.41666pt}\pgfsys@lineto{0.0pt}{3.57222pt }\pgfsys@lineto{3.19446pt}{3.57222pt}\pgfsys@lineto{3.19446pt}{-0.41666pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.19444pt\hbox {\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to2.24pt{\vbox to2.79pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt }\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox {\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to1.6pt{\vbox to1.99pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.20833pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.20833pt}\pgfsys@lineto{0.0pt}{1.7861pt} \pgfsys@lineto{1.59723pt}{1.7861pt}\pgfsys@lineto{1.59723pt}{-0.20833pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-1.59721pt\hbox {\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}0}}\left[\sum_{i\neq j }\log\frac{R_{t}(i,j)}{R_{T}(i,j)}\sum_{0<u\leq t}H_{u}^{ij}\Bigg{|}X_{t}=i \right]\right]$ $$

On the left side, to compute the expected value, we would need to sample the entire trajectory $(X_{u})_{u\in[T,0]}$ and then calculate the inner sum. On the right side, this process is broken down into two steps: first, we sample a trajectory from $X_{T}$ to $X_{t}$, and then, given $X_{t}$, we sample from $X_{t}$ to $X_{0}$ to compute the inner expectation. Our main insight is that the expected value in the second step can be obtained in closed form.

The term $\sum_{0<u\leq t}H_{u}^{ij}$ counts the number of transitions from state $i$ to state $j$ in the interval $[t,0)$. Since $\mathbb{Q}_{\text{paths}}^{t\mathrel{\mathchoice{\mkern 2.0mu\leavevmode
\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{\pgfpicture
\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{
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{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to
0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt}
\pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt}
\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox
{\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
\leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{
\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{
\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}
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{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to
0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt}
\pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt}
\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox
{\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
\leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.13pt{\vbox to3.91pt{
\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.40833pt\hbox to0.0pt{
\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}
{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to
0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.40833pt}\pgfsys@lineto{0.0pt}{3.50076pt
}\pgfsys@lineto{3.13055pt}{3.50076pt}\pgfsys@lineto{3.13055pt}{-0.40833pt}
\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.13055pt\hbox
{\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
\leavevmode\leavevmode\hbox{ \leavevmode\hbox to2.24pt{\vbox to2.79pt{
\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{
\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}
{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to
0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt
}\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt}
\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox
{\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}0}$ uses a single rate matrix over this interval, for $i\neq j$, the expected number of transitions from $i$ to $j$ is approximately given by
(Campbell et al., , Sec. 4.3):

$$ $\mathbb{E}_{\mathbb{Q}_{\text{paths}}^{t\mathrel{\mathchoice{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.19pt{\vbox to3.99pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.41666pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.41666pt}\pgfsys@lineto{0.0pt}{3.57222pt }\pgfsys@lineto{3.19446pt}{3.57222pt}\pgfsys@lineto{3.19446pt}{-0.41666pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.19444pt\hbox {\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.19pt{\vbox to3.99pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.41666pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.41666pt}\pgfsys@lineto{0.0pt}{3.57222pt }\pgfsys@lineto{3.19446pt}{3.57222pt}\pgfsys@lineto{3.19446pt}{-0.41666pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.19444pt\hbox {\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to2.24pt{\vbox to2.79pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt }\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox {\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to1.6pt{\vbox to1.99pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.20833pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.20833pt}\pgfsys@lineto{0.0pt}{1.7861pt} \pgfsys@lineto{1.59723pt}{1.7861pt}\pgfsys@lineto{1.59723pt}{-0.20833pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-1.59721pt\hbox {\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}0}}\left[\sum_{0<u \leq t}H_{u}^{ij}\right]\approx R_{t}(i,j)\Delta t$ $$

where $\Delta t$ represents the size of the time interval.

Using this result, we can simplify Equation Eq. (13) as follows (Derivation in Appendix A.5):

$$ $\textrm{KLUB}_{\mathbb{Q}_{T}}\big{(}\mathbb{Q}_{0}^{T\mathrel{\mathchoice{ \mkern 2.0mu\leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to 5.58pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt {\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.13pt{\vbox to3.91pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.40833pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.40833pt}\pgfsys@lineto{0.0pt}{3.50076pt }\pgfsys@lineto{3.13055pt}{3.50076pt}\pgfsys@lineto{3.13055pt}{-0.40833pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.13055pt\hbox {\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to2.24pt{\vbox to2.79pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt }\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox {\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}t\mathrel{ \mathchoice{\mkern 2.0mu\leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt {\vbox to5.58pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt \hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{ rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ } \pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt} \pgfsys@invoke{ }\nullfont\hbox to0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333 pt}\pgfsys@lineto{0.0pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{-0.58333pt}\pgfsys@closepath\pgfsys@clipnext \pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}}{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.13pt{\vbox to3.91pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.40833pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.40833pt}\pgfsys@lineto{0.0pt}{3.50076pt }\pgfsys@lineto{3.13055pt}{3.50076pt}\pgfsys@lineto{3.13055pt}{-0.40833pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.13055pt\hbox {\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to2.24pt{\vbox to2.79pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt }\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox {\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}0}\,\big{\|}\, \mathbb{Q}_{0}^{T\mathrel{\mathchoice{\mkern 2.0mu\leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{\pgfpicture\makeatletter\hbox{\hskip 0 .0pt\lower-0.58333pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0} \pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ } \pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{{}{}{}{} \pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt}\pgfsys@lineto {4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt}\pgfsys@closepath \pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}}{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.13pt{\vbox to3.91pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.40833pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.40833pt}\pgfsys@lineto{0.0pt}{3.50076pt }\pgfsys@lineto{3.13055pt}{3.50076pt}\pgfsys@lineto{3.13055pt}{-0.40833pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.13055pt\hbox {\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to2.24pt{\vbox to2.79pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt }\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox {\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}0}\big{)}\approx \mathbb{E}_{\mathbb{Q}_{\text{paths}}^{T\mathrel{\mathchoice{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.19pt{\vbox to3.99pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.41666pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.41666pt}\pgfsys@lineto{0.0pt}{3.57222pt }\pgfsys@lineto{3.19446pt}{3.57222pt}\pgfsys@lineto{3.19446pt}{-0.41666pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.19444pt\hbox {\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.19pt{\vbox to3.99pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.41666pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.41666pt}\pgfsys@lineto{0.0pt}{3.57222pt }\pgfsys@lineto{3.19446pt}{3.57222pt}\pgfsys@lineto{3.19446pt}{-0.41666pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ 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\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-1.59721pt\hbox {\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}t}}\left[\sum_{j\neq X _{t}}\log\frac{R_{t}(X_{t},j)}{R_{T}(X_{t},j)}\times R_{t}(X_{t},j)\Delta t \right].$ (14) $$

Figure: Figure 4: The values of KLUB with respect to $t$. Blue lines show estimated results from individual $(X_{t})_{t\in[T,0]}$, while the red line is the average.
Refer to caption: x3.png

In Eq. (13), obtaining a reliable KLUB estimation requires sampling trajectories to capture transitions between various states $i$ and $j$, which is both inaccurate and sample-inefficient—especially when dealing with large state spaces. For instance, in text generation tasks where the state space can be around 50,257, it’s practically impossible to estimate the transition ratios between all pairs $(i,j)$ through sampling alone. In contrast, Eq. (14) allows us to compute this component in closed form, leading to more reliable and efficient calculations. The full algorithm for KLUB computation, combining Techniques 1 and 2, is provided in Appendix B.1.

Now, we can maximize Eq. (14) using a standard optimization algorithm. Before selecting the algorithm, we conduct a preliminary check to observe how the KLUB value changes with respect to $t$ (Figure 4). Interestingly, it exhibits a unimodal shape. Since we are working with a single variable $t$ in the unimodal optimization landscape, we use the golden section search (Press, ), a well-known one-dimensional search algorithm, to find the value of $t$ that maximizes KLUB. This method has the advantage of not relying on hyperparameters like learning rate, which can significantly affect performance.

### 4 Experiments

In this section, we evaluate the Jump Your Steps (JYS) sampling schedule across various datasets and models. We compare the JYS schedule with the uniform sampling schedule, which sets all intervals to the same size. Except for the Countdown dataset, we use open-sourced pretrained models for our experiments. It is important to note that the Gillespie algorithm is only applicable to an absorbing transition matrix, as uniform or Gaussian transition kernels do not have a fixed number of transitions. For further experimental details and additional qualitative results, please refer to the Appendix C and D.2.

Figure: Figure 5: Performance comparisons on Countdown. The JYS schedule enhances sampling quality across different types of samplers.
Refer to caption: x4.png

#### 4.1 The CountDown Dataset

Following Zhao et al. (), to evaluate our sampling schedule performance, we created a synthetic sequence dataset with a strong position-wise correlation structure. Each sample consists of 256 tokens, and each token has a value between 0 and 31. Each data sequence $X^{0:255}$ is generated according to the following rules:

$$ $\displaystyle X^{0}$ $\displaystyle\sim\text{Uniform}\{1,\ldots,S\},$ $\displaystyle X^{d+1}\mid X^{d}$ $\displaystyle\sim\begin{cases}\delta_{X^{d}-1}&\text{if }X^{d}\neq 0\\ \text{Uniform}\{1,\ldots,S\}&\text{if }X^{d}=0\end{cases}$ $$

We trained a SEDD (Lou et al., ) with an absorb transition matrix on this generated data. We measure the model performance by the proportion of generated samples that violated the rule, i.e., failed to count downwards from the previous token. The results are shown in Figure 5. We observe that the JYS schedule has fewer errors compared to the uniform schedule for the same NFE.

#### 4.2 CIFAR-10

Figure: Figure 6: (Left) Performance comparisons on CIFAR-10. (Right) Samples generated using the uniform and JYS schedules, both with NFE 64.
Refer to caption: x5.png

We demonstrate our sampling schedule in the image domain. For this experiment, we use a pretrained model from Campbell et al. (), which employs a gaussian transition matrix and denoising parameterization. Each data sample is a flattened image with a length of $3\times 32\times 32$, composed of tokens with values ranging from 0 to 255.

Figure 6 (left) shows the FID score using 50k samples with the number of function evaluations (NFE) from 32 to 256. We observe that, for all NFEs, the JYS schedule yields a better FID score at the same NFE compared to the uniform schedule. Figure 6 (right) shows randomly generated unconditional CIFAR-10 samples with NFE = 64. The uniform schedule produces blurry images, whereas the images generated using the JYS schedule exhibit clearer colors and shapes of objects.

#### 4.3 Monophonic Music

We test our method on conditional music generation using the Lakh pianoroll dataset (Raffel, ; Dong et al., ). For this experiment, we employ a pretrained model from Campbell et al. (), which uses a uniform transition matrix and denoising parameterization. Each data sequence contains 256 timesteps (16 per bar), and we measure performance by conditioning on two bars to generate the remaining 14 bars, following the setup in Campbell et al. ().

We evaluate how different the generated results were when using a smaller NFE (from 2 to 64), compare to samples generated with an NFE of 512. Specifically, we calculate the Hellinger distance between the note distributions in the generated samples. The results are presented in Figure 8.
Given the same NFE, we observe that samples generated using our method were more similar to those generated with a high NFE.

Figure: Figure 7: Performance comparisons on Monophonic music.
Refer to caption: x6.png

#### 4.4 Text modeling

Finally, we validate our method on text generation. For this experiment, we use a pretrained model from Lou et al. (), which employs an absorbing transition matrix and score-based parameterization. We use two model sizes, SEDD-small and SEDD-medium, in the experiments; both models use the GPT-2 tokenizer and were trained on OpenWebText. The JYS schedule, optimized on SEDD-small, is also used for the experiments with SEDD-medium.

Following Lou et al. (), we measure the generative perplexity of sampled sequences (using a GPT-2 large for evaluation). We generated 1,024 samples and each sample constructed with sequences of 1,024 tokens. We simulate 16 to 256 NFE for generation. Figure 8 shows the results, demonstrating better perplexity at the same NFE.

Figure: Figure 9: Sampling trajectories for different dataset-transition matrix combinations. (Top) Trajectories using infinitesimal timesteps. (Middle) Trajectories using the JYS schedule. (Bottom) Optimized sampling schedule with JYS.
Refer to caption: x8.png

#### 4.5 Characteristics of Jump-Your-Step Sampling schedule

In Section 3.1, we hypothesized that in regions where the conditional mutual information is low, the CDE would also be small, allowing steps to be skipped with minimal performance degradation. Here, we aim to verify if the JYS operates as expected according to this hypothesis.

Figure 9 shows the JYS sampling schedules optimized for various transition matrices. First, in the absorb case (Left), as discussed in Figure 1 (Bottom), we observe that large intervals are concentrated toward the latter part of the process. This occurs because the previously generated tokens help reduce the uncertainty of other tokens. In contrast, in the uniform case (Middle), large intervals appear at the beginning. This can be understood as a result of $X_{t}$ following a uniform distribution, making the tokens independent, leading to lower mutual information for larger $t$. Lastly, for the Gaussian transition (Right), large intervals appear initially and then increase again over time. This pattern suggests that, initially, tokens behave independently like in the uniform case, but after a certain timestep, the effect of resolving uncertainty, similar to the absorb case, becomes more significant as more tokens are generated.

These observations demonstrate how the JYS schedule adapts to the underlying data distribution and token dependencies, effectively allocating computational resources where they are most needed to minimize the compounding error during parallel sampling.

### 5 Related Work

#### 5.1 Efficient Sampling for Continuous Diffusion Models

After the seminal work by Song et al. (), which interpreted diffusion models as a Stochastic or Ordinary Differential Equations (SDE/ODE), various SDE (Jolicoeur-Martineau et al., ; Xu et al., ) and ODE solvers (Song et al., ; Lu et al., ; Zhang & Chen, ; Dockhorn et al., ; Liu et al., ; Zheng et al., ) have been proposed to improve sampling speed.

The work most closely related to ours is “Align Your Step” (Sabour et al., ), which focuses on sampling schedule optimization in continuous diffusion models. In contrast, our approach targets discrete diffusion models, where we derive the KLUB for CTMCs to optimize the sampling schedule. We also propose a computationally efficient algorithm for KLUB computation and optimization.

#### 5.2 Discrete Diffusion Models

Several approaches have been developed for training discrete diffusion models, including denoising parameterization (Austin et al., ; Campbell et al., ; Gu et al., ; Gat et al., ; Campbell et al., ; Shi et al., ) and score parameterization (Sun et al., ; Meng et al., ; Lou et al., ). Recently, SEDD has outperformed GPT-2 in text modeling, gaining traction as an alternative to autoregressive models (Deschenaux & Gulcehre, ).

In terms of sampling, two main directions have emerged. The first focuses on efficient sampling, with $\tau$-leaping for CTMC and methods like analytic sampling (Sun et al., ), Tweedie sampling (Lou et al., ), and $k$-Gillespie (Zhao et al., ) improving accuracy. The second aims to reduce compounding error via corrector steps, such as random correctors (Campbell et al., ), separate corrector training (Lezama et al., ), or enabling the model to act as an informed corrector (Zhao et al., ).

These methods complement the sampling schedule optimization explored in this paper and can be used together for further improvements.

### 6 Conclusions

We present Jump Your Steps, a principled method designed to optimize the sampling schedule and minimize these numerical errors without incurring additional computational costs during inference. Unlike existing approaches that rely on extra computational efforts, such as predictor-corrector methods, our technique operates independently and efficiently. Through extensive evaluations on synthetic and real-world datasets—including monophonic piano, image, and text generation—we demonstrate that our method consistently enhances performance across different transition kernels in discrete diffusion models and effectively complements various samplers.

## Appendix

### Appendix A Theoretical Details

#### A.1 Proof of Theorem 3.1

Although we treated the two-dimensional case where we have $X_{t}=(X^{1}_{t},X^{2}_{t})$ in the main body,
we consider the general $d$-dimensional case with $X_{t}=(X_{t}^{1},\ldots,X_{t}^{d})$.
In that case, the definition of $\mathcal{E}_{\mathrm{CDE}}$ is given by

$$ $\mathcal{E}_{\mathrm{CDE}}(s\to t|\mathbf{x}_{s})\triangleq\mathcal{D}_{ \mathrm{KL}}(P_{\mathbf{x}_{t}|\mathbf{x}_{s}}\|P_{X_{t}^{1}|\mathbf{x}_{s}} \otimes\cdots\otimes P_{X_{t}^{d}|\mathbf{x}_{s}}).$ (15) $$

In general, for a discrete probability distribution $\mathbb{P}_{\mathrm{prior}}(\cdot)$
over the space $\mathcal{S}$
and a conditional distribution (or denoiser) $\mathbb{P}_{\mathrm{cond}}(\cdot|\cdot)$ over the same space,
i,e, $\mathbb{P}_{\mathrm{cond}}:\mathcal{S}\times\mathcal{S}\to\mathbb{R}$,
let us just write the resulting distribution as follows:

$$ $\mathbb{P}_{\mathrm{cond}}\mathbb{P}_{\mathrm{prior}}\triangleq\mathbb{E}_{ \mathbf{x}\sim\mathbb{P}_{\mathrm{prior}}}[\mathbb{P}_{\mathrm{cond}}(\cdot| \mathbf{x})]=\sum_{x\in\mathcal{S}}\mathbb{P}_{\mathrm{cond}}(\cdot|x)\mathbb{ P}_{\mathrm{prior}}(x).$ $$

Then, if we define $\mathbb{P}_{t_{i+1}|t_{i}}(\cdot|\mathbf{x}_{t_{i}})\triangleq P_{X_{t_{i+1}}|
\mathbf{x}_{t_{i}}}$
and $\mathbb{Q}_{t_{i+1}|t_{i}}(\cdot|\mathbf{x}_{t_{i}})\triangleq P_{X_{t}^{1}|
\mathbf{x}_{s}}\otimes\cdots\otimes P_{X_{t}^{d}|\mathbf{x}_{s}}$
following equation 15,
we can denote the target distributions in the theorem as follows:

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{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}0}=\mathbb{Q}_{t_{N} |t_{N-1}}\cdots\mathbb{Q}_{t_{1}|t_{0}}\mathbb{P}_{t_{0}}.$ $$

For simplicity,
let us also define the mid-time distributions as

$$ $\mathbb{P}_{t_{i}}\triangleq\mathbb{P}_{t_{i}|t_{i-1}}\cdots\mathbb{P}_{t_{1}| t_{0}}\mathbb{P}_{t_{0}},\qquad\mathbb{Q}_{t_{i}}\triangleq\mathbb{Q}_{t_{i}|t _{i-1}}\cdots\mathbb{Q}_{t_{1}|t_{0}}\mathbb{P}_{t_{0}},\qquad\mathbb{Q}_{t_{0 }}\triangleq\mathbb{P}_{t_{0}}.$ $$

Let us now consider the case where $s=t_{i-1}$ and $t=t_{i}$ for some $i>0$
specifically.
Let

$$ $\mathbb{P}_{t,s}(y,x)\triangleq\mathbb{P}_{t|s}(y|x)\mathbb{P}_{s}(x),\qquad \mathbb{Q}_{t,s}(y,x)\triangleq\mathbb{Q}_{t|s}(y|x)\mathbb{Q}_{s}(x)$ $$

to denote joint distributions on $\mathcal{S}\times\mathcal{S}$.
Then, since $\mathbb{P}_{t}=\mathbb{P}_{t|s}\mathbb{P}_{s}$ and $\mathbb{Q}_{t}=\mathbb{Q}_{t|s}\mathbb{Q}_{s}$
are marginal distributions of $\mathbb{P}_{t,s}$ and $\mathbb{Q}_{t,s}$, we have

$$ $\displaystyle\mathcal{D}_{\mathrm{KL}}(\mathbb{P}_{t}\|\mathbb{Q}_{t})$ $\displaystyle\leq\mathcal{D}_{\mathrm{KL}}(\mathbb{P}_{t,s}\|\mathbb{Q}_{t,s})$ $\displaystyle=\sum_{y,x}\mathbb{P}_{t|s}(y|x)\mathbb{P}_{s}(x)\log\frac{ \mathbb{P}_{t|s}(y|x)\mathbb{P}_{s}(x)}{\mathbb{Q}_{t|s}(y|x)\mathbb{Q}_{s}(x)}$ $\displaystyle=\sum_{y,x}\mathbb{P}_{t|s}(y|x)\mathbb{P}_{s}(x)\log\frac{ \mathbb{P}_{s}(x)}{\mathbb{Q}_{s}(x)}+\sum_{y,x}\mathbb{P}_{t|s}(y|x)\mathbb{P }_{s}(x)\log\frac{\mathbb{P}_{t|s}(y|x)}{\mathbb{Q}_{t|s}(y|x)}$ $\displaystyle=\mathcal{D}_{\mathrm{KL}}(\mathbb{P}_{s}\|\mathbb{Q}_{s})+\sum_{ x}\mathbb{P}_{s}(x)\mathcal{D}_{\mathrm{KL}}(\mathbb{P}_{t|s}(\cdot|x)\| \mathbb{Q}_{t|s}(\cdot|x))$ $\displaystyle=\mathcal{D}_{\mathrm{KL}}(\mathbb{P}_{s}\|\mathbb{Q}_{s})+ \mathbb{E}_{\mathbf{x}_{s}\sim\mathbb{P}_{s}}[\mathcal{E}_{\mathrm{CDE}}(s\to t |\mathbf{x}_{s})]$ $\displaystyle=\mathcal{D}_{\mathrm{KL}}(\mathbb{P}_{s}\|\mathbb{Q}_{s})+ \mathcal{E}_{\mathrm{CDE}}(s\to t).$ (16) $$

By iteratively using equation 16,
we obtain

$$ $\displaystyle\mathcal{D}_{\mathrm{KL}}(\mathbb{P}_{0}\|\mathbb{Q}_{0})$ $\displaystyle=\mathcal{D}_{\mathrm{KL}}(\mathbb{P}_{t_{N}}\|\mathbb{Q}_{t_{N}})$ $\displaystyle\leq\mathcal{D}_{\mathrm{KL}}(\mathbb{P}_{t_{N-1}}\|\mathbb{Q}_{t _{N-1}})+\mathcal{E}_{\mathrm{CDE}}(t_{N-1}\to t_{N})$ $\displaystyle\leq\mathcal{D}_{\mathrm{KL}}(\mathbb{P}_{t_{N-2}}\|\mathbb{Q}_{t _{N-2}})+\mathcal{E}_{\mathrm{CDE}}(t_{N-2}\to t_{N-1})+\mathcal{E}_{\mathrm{ CDE}}(t_{N-1}\to t_{N})$ $\displaystyle\ \ \vdots$ $\displaystyle\leq\mathcal{D}_{\mathrm{KL}}(\mathbb{P}_{t_{0}}\|\mathbb{Q}_{t_{ 0}})+\sum_{i=0}^{N-1}\mathcal{E}_{\mathrm{CDE}}(t_{i}\to t_{i+1}).$ $$

Since we set $\mathbb{P}_{t_{0}}=\mathbb{Q}_{t_{0}}$,
we have completed the proof.

#### A.2 Proof of Theorem 3.2

Although we state the theorem for a backward CTMC from time $s$ to time $t$ ($t<s$)
and bound the KL-divergence at time $t$,
here we will just consider the forward CTMC from time 0 0 to time $T$ and bound the KL-divergence at time $T$ for simplicity.
After this change of the time direction and interval,
the formal statement and proof of Theorem 3.2 is given in Theroem A.3.

To derive the KL-divergence upper bound (KLUB) for continuous-time Markov chains (CTMCs), we first adopt the change of measure $\frac{d\mathbb{P}_{\mathrm{paths}}}{d\mathbb{Q}_{\mathrm{paths}}}$ for CTMCs from Section 3 of Ding & Ning (). Next, we compute and organize the equation for the $D_{\mathrm{KL}}({\mathbb{P}_{\mathrm{paths}}|\mathbb{Q}_{\mathrm{paths}}})=
\mathbb{E}_{\mathbb{P}_{\mathrm{paths}}}\left[\log\frac{d\mathbb{P}_{\mathrm{
paths}}}{d\mathbb{Q}_{\mathrm{paths}}}\right]$.

We consider the following two forward CTMCs over $[0,T]$:

$$ $\begin{cases}\textrm{CTMC 1}:&q^{1}_{u+du|u}(y\mid x)=\delta_{xy}+R^{1}_{t}(x, y)du+o(du),\\ \textrm{CTMC 2}:&q^{2}_{u+du|u}(y\mid x)=\delta_{xy}+R^{2}_{t}(x,y)du+o(du). \end{cases}$ $$

Here, $R^{1}_{t}$ and $R^{2}_{t}$ represent the rate matrices of each CTMC, with a finite state space $S=\{x_{1},\cdots,x_{N}\}$,
and $du>0$.

We introduce some notations. Define the functions $H_{t}^{i},H_{t}^{ij}$ as follows:

$$ $H_{t}^{i}:=\delta(X_{t}-x_{i}),\quad H_{t}^{ij}:=H_{t}^{j}H_{t-}^{i},$ $$

where $\delta(\cdot)$ denotes the Dirac delta function, and $t-$ is the left limit of $t$. By definition, $H_{t}^{ij}=1$ indicates a transition from $x_{i}$ to $x_{j}$ at time $t$. The CTMC $(X_{t})_{t\in[0,T]}$ is defined as a function from the sample space $\Omega$ to the path space $\mathcal{C}\triangleq[0,T]\times S$, i.e., $X:\Omega\rightarrow\mathcal{C}$.

We define two probability measures over the path space:

- •
$\mathbb{\mathbb{P}}_{\mathrm{paths}}$, under which $(X^{1}_{t})_{t\in[0,T]}$ has the law of CTMC 1.
- •
$\mathbb{\mathbb{Q}}_{\mathrm{paths}}$, under which $(X^{2}_{t})_{t\in[0,T]}$ has the law of CTMC 2.

First, we adopt the change of measure for CTMCs from Ding & Ning ().

###### Proposition A.1 .

(Ding & Ning (), Eq. 3.2)
Consider a family of bounded real-valued processes $\{\kappa_{t}(i,j)\}_{i,j\in\{1,\cdots,N\}}$, such that $\kappa_{t}(i,j)>-1$ and $\kappa_{ii}(t)=0$. Define $(\eta_{t})_{0\leq t\leq T}$ as

$$ $\eta_{t}=e^{-L_{t}}\prod_{0<u\leq T}\left(1+\sum_{i,j=1}^{N}\kappa_{u}(i,j)H_{ u}^{ij}\right),$ $$

where $L_{t}=\int^{t}_{0}\sum_{i,j=1}^{N}\kappa_{u}(i,j)R^{1}_{u}(i,j)H^{i}_{u}du$. This result implies the existence of a probability measure $\mathbb{Q}$ defined by

$$ $\frac{d\mathbb{Q}}{d\mathbb{P}}=\eta_{t}.$ (17) $$

The following result allows us to define $\kappa_{t}(i,j)$ for two CTMCs.

###### Proposition A.2 .

(Ding & Ning (), Eq. 3.4)
For the probability measure $\mathbb{Q}$ defined in Eq.(17), if $(X^{1}_{t})_{t\in[0,T]}$ is a CTMC under $\mathbb{P}$ with rate matrix $R^{1}$ and $(X^{2}_{t})_{t\in[0,T]}$ is a CTMC under $\mathbb{Q}$ with rate matrix $R^{2}$, then $R^{2}$ satisfies

$$ $\displaystyle R^{2}_{ii}(t)$ $\displaystyle=-\sum_{j\neq i}R^{2}_{t}(i,j),$ $\displaystyle R^{2}_{t}(i,j)$ $\displaystyle=(1+\kappa_{t}(i,j))R^{1}_{t}(i,j).$ $$

From Proposition A.2, defining $\kappa_{t}(i,j)=R^{2}_{t}(i,j)/R^{1}_{t}(i,j)-1$ (as in Proposition A.1), we can compute the change of measure between the two CTMC-defined measures $\mathbb{P}_{\mathrm{paths}}$ and $\mathbb{Q}_{\mathrm{paths}}$ as $\frac{d\mathbb{P}_{\mathrm{paths}}}{d\mathbb{Q}_{\mathrm{paths}}}$.

Now, we are ready to derive the KLUB for CTMCs:

###### Theorem A.3 .

(KL-divergence Upper bound, KLUB) Consider the following two forward CTMCs:

$$ $\begin{cases}\textrm{CTMC 1}:&q^{1}_{u+du|u}(y\mid x)=\delta_{xy}+R^{1}_{t}(x, y)du+o(du),\\ \textrm{CTMC 2}:&q^{2}_{u+du|u}(y\mid x)=\delta_{xy}+R^{2}_{t}(x,y)du+o(du). \end{cases}$ $$

Here, $R^{1}_{t}$ and $R^{2}_{t}$ represent the rate matrices of each CTMC over $[0,T]$, with a finite state space $S=\{x_{1},\cdots,x_{N}\}$.
Let $\mathbb{P}_{T}$ and $\mathbb{Q}_{T}$ be the resulting probability distributions at the time $T$ of the outputs of CTMC 1 and 2, respectively. Then we have:

$$ $D_{\mathrm{KL}}(\mathbb{P}_{T}\|\mathbb{Q}_{T})\leq\mathbb{E}_{\mathbb{P}_{ \mathrm{paths}}}\left[\sum_{\begin{subarray}{c}H_{u}^{ij}=1\\ 0<u\leq T\end{subarray}}\log\frac{R^{1}_{u}(i,j)}{R^{2}_{u}(i,j)}.\right]$ $$

where, $\mathbb{P}_{\mathrm{paths}}$ refers to the distribution over path space $(X_{t})_{t\in[0,T]}\in[0,T]\times S$ generated by running CTMC 1, and $H_{u}^{ij}=1$ represent the transition of state from $x_{i}$ to $x_{j}$ at time $u$.

###### Proof.

This result is derived by combining Proposition A.1 and A.2 and rearranging the resulting expression.

$$ $\displaystyle D_{\mathrm{KL}}\left(\mathbb{P}_{\mathrm{paths}}\|\mathbb{Q}_{ \mathrm{paths}}\right)$ $\displaystyle=\mathbb{E}_{\mathbb{P}_{\mathrm{paths}}}\left[\log\frac{d\mathbb {P}_{\mathrm{paths}}}{d\mathbb{Q}_{\mathrm{paths}}}\right]$ (18a) $\displaystyle=\mathbb{E}_{\mathbb{P}_{\mathrm{paths}}}\left[\log\eta_{t}^{-1}\right]$ (18b) $\displaystyle=\mathbb{E}_{\mathbb{P}_{\mathrm{paths}}}\left[\int_{0}^{t}\sum_{ i,j=1}^{N}\kappa_{u}(i,j)R^{1}_{u}(i,j)H^{i}_{u}du-\log\prod_{0<u\leq T}(1+ \sum_{i,j=1}^{N}\kappa_{u}(i,j)H_{u}^{ij})\right]$ (18c) $\displaystyle=\mathbb{E}_{\mathbb{P}_{\mathrm{paths}}}\left[\int_{0}^{t}\sum_{ i,j=1}^{N}(R^{2}_{u}(i,j)-R^{1}_{u}(i,j))H^{i}_{u}du+\sum_{\begin{subarray}{c} H_{u}^{ij}=1\\ 0<u\leq T\end{subarray}}(\log R^{1}_{u}(i,j)-\log R^{2}_{u}(i,j))\right]$ (18d) $\displaystyle=\mathbb{E}_{\mathbb{P}_{\mathrm{paths}}}\left[\sum_{\begin{ subarray}{c}H_{u}^{ij}=1\\ 0<u\leq T\end{subarray}}(\log R^{1}_{u}(i,j)-\log R^{2}_{u}(i,j))\right].$ (18e) $$

Step (18c) follows from Proposition A.1, while step (18d) is derived by substituting $\kappa_{t}(i,j)=\frac{R^{2}_{t}(i,j)}{R^{1}_{t}(i,j)}-1$ as defined in Proposition A.2. Additionally, step (18e) utilizes the property of the rate matrix, where $\sum_{j}R^{1}_{t}(i,j)=0$ for any $t$ and $i$.

Finally, we denote the distributions generated by CTMC 1 and CTMC 2 at time $T$ as $\mathbb{P}_{T}$ and $\mathbb{Q}_{T}$, respectively. Since $\mathbb{P}_{T}$ and $\mathbb{Q}_{T}$ are the marginal distributions of $\mathbb{P}_{\mathrm{paths}}$ and $\mathbb{Q}_{\mathrm{paths}}$ at time $T$, by the data processing inequality, the KL-divergence $D_{\mathrm{KL}}(\mathbb{P}_{T}\|\mathbb{Q}_{T})$ is upper-bounded by $D_{\mathrm{KL}}(\mathbb{P}_{\mathrm{paths}}\|\mathbb{Q}_{\mathrm{paths}})$, concluding the proof.
∎

It is important to note that the KLUB derived here for CTMCs does not fully capture the case of discrete diffusion models, where the reverse rate matrix depends on the state. Nonetheless, we believe the metric derived here provides a useful proxy for estimating the error introduced when using $\tau$-leaping to sample in discrete diffusion models. Finding KLUB for state-dependent CTMCs would be an interesting direction for future work.

###### Proposition A.1 .

###### Proposition A.2 .

###### Theorem A.3 .

###### Proof.

#### A.3 Proof of Eq. ( 9 )

Let $X^{i}$ refers to the $i$-th dimension of $X$. Define $\mathbb{P}_{t_{i+1}|t_{i}}(\cdot|\mathbf{x}_{t_{i}})\triangleq P_{X_{t_{i+1}}|
\mathbf{x}_{t_{i}}}$
and $\mathbb{Q}_{t_{i+1}|t_{i}}(\cdot|\mathbf{x}_{t_{i}})\triangleq P_{X_{t}^{1}|
\mathbf{x}_{s}}\otimes\cdots\otimes P_{X_{t}^{d}|\mathbf{x}_{s}}$ We can now derive the following relationship:

$$ $\mathcal{E}_{\mathrm{CDE}}(s\shortrightarrow t\,|\,\mathbf{x}_{s})\triangleq D _{KL}\left(\mathbb{P}_{t|s}\big{\|}\mathbb{Q}_{t|s}\right)\leq D_{KL}\left( \mathbb{P}_{[s,t]|s}\big{\|}\mathbb{Q}_{[s,t]|s}\right)=\text{KLUB}_{\mathbf{x }_{s}}(\mathbb{P}_{t}\|\mathbb{Q}_{t}).$ $$

where $\text{KLUB}_{\mathbf{x}_{s}}$ represents comparing two continuous-time Markov chains (CTMCs), both starting at the initial point $\mathbf{x}_{s}$.

Now we are ready to prove the result:

$$ $\displaystyle\mathcal{E}_{\mathrm{CDE}}(s\shortrightarrow t)$ $\displaystyle\triangleq\mathbb{E}_{\mathbf{X}_{s}\sim\mathbb{P}_{s}}\left[ \mathcal{E}_{\mathrm{CDE}}(s\shortrightarrow t\,|\,\mathbf{X}_{s})\right]$ $\displaystyle\leq\mathbb{E}_{\mathbf{X}_{s}\sim\mathbb{P}_{s}}\left[\textrm{{ KLUB}}_{\mathbf{X}_{s}}(\mathbb{P}_{t}\|\mathbb{Q}_{t}^{s\mathrel{\mathchoice{ \mkern 2.0mu\leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to 5.58pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt {\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.13pt{\vbox to3.91pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.40833pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.40833pt}\pgfsys@lineto{0.0pt}{3.50076pt }\pgfsys@lineto{3.13055pt}{3.50076pt}\pgfsys@lineto{3.13055pt}{-0.40833pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.13055pt\hbox {\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to2.24pt{\vbox to2.79pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt }\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox {\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}t})\right]$ $\displaystyle=\mathbb{E}_{\mathbf{X}_{s}\sim\mathbb{P}_{s}}\left[\mathbb{E}_{ \mathbb{P}_{t|s}}\left[\sum_{\begin{subarray}{c}H_{u}^{ij}=1\\ 0<u\leq T\end{subarray}}\log\frac{R^{1}_{u}(i,j)}{R^{2}_{u}(i,j)}\right]\right]$ $\displaystyle=\mathbb{E}_{\mathbb{P}_{\mathrm{paths}}}\left[\sum_{\begin{ subarray}{c}H_{u}^{ij}=1\\ 0<u\leq T\end{subarray}}\log\frac{R^{1}_{u}(i,j)}{R^{2}_{u}(i,j)}\right]$ $\displaystyle=\textrm{{KLUB}}_{\mathbb{P}_{s}}(\mathbb{P}_{t}\|\mathbb{Q}^{s \mathrel{\mathchoice{\mkern 2.0mu\leavevmode\leavevmode\hbox{ \leavevmode\hbox to 4.47pt{\vbox to5.58pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333 pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor }{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ } \pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt} \pgfsys@invoke{ }\nullfont\hbox to0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333 pt}\pgfsys@lineto{0.0pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{-0.58333pt}\pgfsys@closepath\pgfsys@clipnext \pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}}{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.13pt{\vbox to3.91pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.40833pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.40833pt}\pgfsys@lineto{0.0pt}{3.50076pt }\pgfsys@lineto{3.13055pt}{3.50076pt}\pgfsys@lineto{3.13055pt}{-0.40833pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.13055pt\hbox {\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to2.24pt{\vbox to2.79pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt }\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox {\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}t}_{t}).$ $$

#### A.4 Technique 1

The approximation is made as follows:

$$ $\displaystyle\mathcal{D}_{\mathrm{KL}}(\mathbb{P}_{path}\|\mathbb{Q}_{path}^{T \mathrel{\mathchoice{\mkern 2.0mu\leavevmode\leavevmode\hbox{ \leavevmode\hbox to 4.47pt{\vbox to5.58pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333 pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor }{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ } \pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt} \pgfsys@invoke{ }\nullfont\hbox to0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333 pt}\pgfsys@lineto{0.0pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{-0.58333pt}\pgfsys@closepath\pgfsys@clipnext \pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}}{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.13pt{\vbox to3.91pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.40833pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.40833pt}\pgfsys@lineto{0.0pt}{3.50076pt }\pgfsys@lineto{3.13055pt}{3.50076pt}\pgfsys@lineto{3.13055pt}{-0.40833pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.13055pt\hbox {\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to2.24pt{\vbox to2.79pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt }\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox {\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}0})-\mathcal{D}_{ \mathrm{KL}}(\mathbb{P}_{path}\|\mathbb{Q}_{path}^{T\mathrel{\mathchoice{ \mkern 2.0mu\leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to 5.58pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt {\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.13pt{\vbox to3.91pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.40833pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.40833pt}\pgfsys@lineto{0.0pt}{3.50076pt }\pgfsys@lineto{3.13055pt}{3.50076pt}\pgfsys@lineto{3.13055pt}{-0.40833pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.13055pt\hbox {\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to2.24pt{\vbox to2.79pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} 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2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.13pt{\vbox to3.91pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.40833pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.40833pt}\pgfsys@lineto{0.0pt}{3.50076pt }\pgfsys@lineto{3.13055pt}{3.50076pt}\pgfsys@lineto{3.13055pt}{-0.40833pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.13055pt\hbox {\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to2.24pt{\vbox to2.79pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt }\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox {\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}0}}\right]$ (19a) $\displaystyle\approx\mathbb{E}_{\mathbb{Q}_{path}^{forward}}\left[\log\frac{ \mathbb{Q}_{path}^{T\mathrel{\mathchoice{\mkern 2.0mu\leavevmode\leavevmode \hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{\pgfpicture\makeatletter\hbox{ \hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0} \pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ } \pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{{}{}{}{} \pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt}\pgfsys@lineto {4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt}\pgfsys@closepath \pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}}{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.13pt{\vbox to3.91pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.40833pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.40833pt}\pgfsys@lineto{0.0pt}{3.50076pt }\pgfsys@lineto{3.13055pt}{3.50076pt}\pgfsys@lineto{3.13055pt}{-0.40833pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.13055pt\hbox {\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to2.24pt{\vbox to2.79pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt }\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox {\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}t\mathrel{ \mathchoice{\mkern 2.0mu\leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt {\vbox to5.58pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt \hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{ rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ } \pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt} \pgfsys@invoke{ }\nullfont\hbox to0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333 pt}\pgfsys@lineto{0.0pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{-0.58333pt}\pgfsys@closepath\pgfsys@clipnext \pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}}{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.13pt{\vbox to3.91pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.40833pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.40833pt}\pgfsys@lineto{0.0pt}{3.50076pt }\pgfsys@lineto{3.13055pt}{3.50076pt}\pgfsys@lineto{3.13055pt}{-0.40833pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.13055pt\hbox {\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to2.24pt{\vbox to2.79pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt }\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox {\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}0}}{\mathbb{Q}_{path }^{T\mathrel{\mathchoice{\mkern 2.0mu\leavevmode\leavevmode\hbox{ \leavevmode \hbox to4.47pt{\vbox to5.58pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower -0.58333pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{ pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ } \pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt} \pgfsys@invoke{ }\nullfont\hbox to0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333 pt}\pgfsys@lineto{0.0pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{-0.58333pt}\pgfsys@closepath\pgfsys@clipnext \pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}}{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.13pt{\vbox to3.91pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.40833pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.40833pt}\pgfsys@lineto{0.0pt}{3.50076pt }\pgfsys@lineto{3.13055pt}{3.50076pt}\pgfsys@lineto{3.13055pt}{-0.40833pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.13055pt\hbox {\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to2.24pt{\vbox to2.79pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt }\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox {\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}0}}\right]$ (19b) $\displaystyle\approx\mathbb{E}_{\mathbb{Q}_{path}^{T\mathrel{\mathchoice{ \mkern 2.0mu\leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.19pt{\vbox to 3.99pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.41666pt\hbox to0.0pt {\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.41666pt}\pgfsys@lineto{0.0pt}{3.57222pt }\pgfsys@lineto{3.19446pt}{3.57222pt}\pgfsys@lineto{3.19446pt}{-0.41666pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.19444pt\hbox {\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.19pt{\vbox to3.99pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.41666pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.41666pt}\pgfsys@lineto{0.0pt}{3.57222pt }\pgfsys@lineto{3.19446pt}{3.57222pt}\pgfsys@lineto{3.19446pt}{-0.41666pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.19444pt\hbox {\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to2.24pt{\vbox to2.79pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt }\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox {\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to1.6pt{\vbox to1.99pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.20833pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 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\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt} \pgfsys@invoke{ }\nullfont\hbox to0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.41666 pt}\pgfsys@lineto{0.0pt}{3.57222pt}\pgfsys@lineto{3.19446pt}{3.57222pt} \pgfsys@lineto{3.19446pt}{-0.41666pt}\pgfsys@closepath\pgfsys@clipnext \pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}}{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.19444pt\hbox {\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.19pt{\vbox to3.99pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.41666pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ 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0.0pt\lower-0.29166pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt }\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox {\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox 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\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}0}}\left[\log\frac{ \mathbb{Q}_{path}^{T\mathrel{\mathchoice{\mkern 2.0mu\leavevmode\leavevmode \hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{\pgfpicture\makeatletter\hbox{ \hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0} \pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ } \pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{{}{}{}{} \pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt}\pgfsys@lineto {4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt}\pgfsys@closepath \pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}}{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.13pt{\vbox to3.91pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.40833pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.40833pt}\pgfsys@lineto{0.0pt}{3.50076pt }\pgfsys@lineto{3.13055pt}{3.50076pt}\pgfsys@lineto{3.13055pt}{-0.40833pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.13055pt\hbox {\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to2.24pt{\vbox to2.79pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt }\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox {\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}t\mathrel{ \mathchoice{\mkern 2.0mu\leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt {\vbox to5.58pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt \hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{ rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ } \pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt} \pgfsys@invoke{ }\nullfont\hbox to0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333 pt}\pgfsys@lineto{0.0pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{-0.58333pt}\pgfsys@closepath\pgfsys@clipnext \pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}}{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ 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2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.13pt{\vbox to3.91pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.40833pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.40833pt}\pgfsys@lineto{0.0pt}{3.50076pt }\pgfsys@lineto{3.13055pt}{3.50076pt}\pgfsys@lineto{3.13055pt}{-0.40833pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.13055pt\hbox {\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } 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{{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.13055pt\hbox {\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to2.24pt{\vbox to2.79pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt }\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt} 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0.0pt\lower -0.58333pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{ pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ } \pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt} \pgfsys@invoke{ }\nullfont\hbox to0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333 pt}\pgfsys@lineto{0.0pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{-0.58333pt}\pgfsys@closepath\pgfsys@clipnext \pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}}{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.13pt{\vbox to3.91pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.40833pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.40833pt}\pgfsys@lineto{0.0pt}{3.50076pt }\pgfsys@lineto{3.13055pt}{3.50076pt}\pgfsys@lineto{3.13055pt}{-0.40833pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.13055pt\hbox {\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to2.24pt{\vbox to2.79pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt }\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox {\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}0})$ (19d) $$

Equation equation 19a assumes that $\mathbb{P}_{path}\approx\mathbb{Q}_{path}^{forward}$ where $\mathbb{Q}_{path}^{forward}$ refers to the distribution made by forward CTMC. In equation equation 19b, we assume that $\mathbb{Q}_{path}\approx\mathbb{Q}_{path}^{T\to t\to 0}$. It is important to note that we can use equation 19a as a formula for KLUB computation, as introduced in Algorithm 1. However, the results of JYS sampling schedule optimization show little difference between the two.

Compared to coarser sampling, KLUB computation can be organized as follows:

$$ $\displaystyle\textrm{{KLUB}}\big{(}\mathbb{Q}_{0}^{T\mathrel{\mathchoice{ \mkern 2.0mu\leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to 5.58pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt {\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.13pt{\vbox to3.91pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.40833pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.40833pt}\pgfsys@lineto{0.0pt}{3.50076pt }\pgfsys@lineto{3.13055pt}{3.50076pt}\pgfsys@lineto{3.13055pt}{-0.40833pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.13055pt\hbox {\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to2.24pt{\vbox to2.79pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt }\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox {\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}t\mathrel{ \mathchoice{\mkern 2.0mu\leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt {\vbox to5.58pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt \hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{ rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ } \pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt} \pgfsys@invoke{ }\nullfont\hbox to0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333 pt}\pgfsys@lineto{0.0pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{-0.58333pt}\pgfsys@closepath\pgfsys@clipnext \pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}}{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.13pt{\vbox to3.91pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.40833pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.40833pt}\pgfsys@lineto{0.0pt}{3.50076pt }\pgfsys@lineto{3.13055pt}{3.50076pt}\pgfsys@lineto{3.13055pt}{-0.40833pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.13055pt\hbox {\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to2.24pt{\vbox to2.79pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt }\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox {\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}0}\,\big{\|}\, \mathbb{Q}_{0}^{T\mathrel{\mathchoice{\mkern 2.0mu\leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{\pgfpicture\makeatletter\hbox{\hskip 0 .0pt\lower-0.58333pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0} \pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ } \pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{{}{}{}{} \pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt}\pgfsys@lineto {4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt}\pgfsys@closepath \pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}}{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.13pt{\vbox to3.91pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.40833pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.40833pt}\pgfsys@lineto{0.0pt}{3.50076pt }\pgfsys@lineto{3.13055pt}{3.50076pt}\pgfsys@lineto{3.13055pt}{-0.40833pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.13055pt\hbox {\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to2.24pt{\vbox to2.79pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt }\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox {\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}0}\big{)}$ $\displaystyle=\mathbb{E}_{\mathbb{Q}_{\mathrm{paths}}^{T\mathrel{\mathchoice{ \mkern 2.0mu\leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.19pt{\vbox to 3.99pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.41666pt\hbox to0.0pt {\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.41666pt}\pgfsys@lineto{0.0pt}{3.57222pt }\pgfsys@lineto{3.19446pt}{3.57222pt}\pgfsys@lineto{3.19446pt}{-0.41666pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ 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{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt }\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox {\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to1.6pt{\vbox to1.99pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.20833pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} 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{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.13pt{\vbox to3.91pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.40833pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.40833pt}\pgfsys@lineto{0.0pt}{3.50076pt }\pgfsys@lineto{3.13055pt}{3.50076pt}\pgfsys@lineto{3.13055pt}{-0.40833pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.13055pt\hbox {\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to2.24pt{\vbox to2.79pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt }\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox {\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}0}_{u}(i,j)}{R^{T \mathrel{\mathchoice{\mkern 2.0mu\leavevmode\leavevmode\hbox{ \leavevmode\hbox to 4.47pt{\vbox to5.58pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333 pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor }{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ } \pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt} \pgfsys@invoke{ }\nullfont\hbox to0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333 pt}\pgfsys@lineto{0.0pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{-0.58333pt}\pgfsys@closepath\pgfsys@clipnext \pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}}{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.13pt{\vbox to3.91pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.40833pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.40833pt}\pgfsys@lineto{0.0pt}{3.50076pt }\pgfsys@lineto{3.13055pt}{3.50076pt}\pgfsys@lineto{3.13055pt}{-0.40833pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.13055pt\hbox {\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to2.24pt{\vbox to2.79pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt }\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox {\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}0}_{u}(i,j)}\right]$ (20a) $\displaystyle=\mathbb{E}_{\mathbb{Q}_{\mathrm{paths}}^{T\mathrel{\mathchoice{ \mkern 2.0mu\leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.19pt{\vbox to 3.99pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.41666pt\hbox to0.0pt {\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.41666pt}\pgfsys@lineto{0.0pt}{3.57222pt }\pgfsys@lineto{3.19446pt}{3.57222pt}\pgfsys@lineto{3.19446pt}{-0.41666pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.19444pt\hbox {\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.19pt{\vbox to3.99pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.41666pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.41666pt}\pgfsys@lineto{0.0pt}{3.57222pt }\pgfsys@lineto{3.19446pt}{3.57222pt}\pgfsys@lineto{3.19446pt}{-0.41666pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.19444pt\hbox {\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to2.24pt{\vbox to2.79pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt }\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox {\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to1.6pt{\vbox to1.99pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.20833pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.20833pt}\pgfsys@lineto{0.0pt}{1.7861pt} \pgfsys@lineto{1.59723pt}{1.7861pt}\pgfsys@lineto{1.59723pt}{-0.20833pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-1.59721pt\hbox {\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}t\mathrel{ \mathchoice{\mkern 2.0mu\leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.19pt {\vbox to3.99pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.41666pt \hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{ rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ } \pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt} \pgfsys@invoke{ }\nullfont\hbox to0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.41666 pt}\pgfsys@lineto{0.0pt}{3.57222pt}\pgfsys@lineto{3.19446pt}{3.57222pt} \pgfsys@lineto{3.19446pt}{-0.41666pt}\pgfsys@closepath\pgfsys@clipnext \pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}}{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.19444pt\hbox {\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.19pt{\vbox to3.99pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.41666pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.41666pt}\pgfsys@lineto{0.0pt}{3.57222pt }\pgfsys@lineto{3.19446pt}{3.57222pt}\pgfsys@lineto{3.19446pt}{-0.41666pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.19444pt\hbox {\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to2.24pt{\vbox to2.79pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt }\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox {\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to1.6pt{\vbox to1.99pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.20833pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.20833pt}\pgfsys@lineto{0.0pt}{1.7861pt} \pgfsys@lineto{1.59723pt}{1.7861pt}\pgfsys@lineto{1.59723pt}{-0.20833pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-1.59721pt\hbox {\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}0}}\left[\sum_{i\neq j }\sum_{u=0}^{t}H_{u}^{ij}\log\frac{R_{t}(i,j)}{R_{T}(i,j)}+\sum_{i\neq j}\sum_ {u=t}^{T}H_{u}^{ij}\cancel{\log\frac{R_{T}(i,j)}{R_{T}(i,j)}}\right]$ (20b) $\displaystyle=\mathbb{E}_{\mathbb{Q}_{\mathrm{paths}}^{T\mathrel{\mathchoice{ \mkern 2.0mu\leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.19pt{\vbox to 3.99pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.41666pt\hbox to0.0pt {\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.41666pt}\pgfsys@lineto{0.0pt}{3.57222pt }\pgfsys@lineto{3.19446pt}{3.57222pt}\pgfsys@lineto{3.19446pt}{-0.41666pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.19444pt\hbox {\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.19pt{\vbox to3.99pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.41666pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.41666pt}\pgfsys@lineto{0.0pt}{3.57222pt }\pgfsys@lineto{3.19446pt}{3.57222pt}\pgfsys@lineto{3.19446pt}{-0.41666pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.19444pt\hbox {\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to2.24pt{\vbox to2.79pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt }\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox {\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to1.6pt{\vbox to1.99pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.20833pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.20833pt}\pgfsys@lineto{0.0pt}{1.7861pt} \pgfsys@lineto{1.59723pt}{1.7861pt}\pgfsys@lineto{1.59723pt}{-0.20833pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-1.59721pt\hbox {\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}t\mathrel{ \mathchoice{\mkern 2.0mu\leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.19pt {\vbox to3.99pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.41666pt \hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{ rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ } \pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt} \pgfsys@invoke{ }\nullfont\hbox to0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.41666 pt}\pgfsys@lineto{0.0pt}{3.57222pt}\pgfsys@lineto{3.19446pt}{3.57222pt} \pgfsys@lineto{3.19446pt}{-0.41666pt}\pgfsys@closepath\pgfsys@clipnext \pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}}{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.19444pt\hbox {\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.19pt{\vbox to3.99pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.41666pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.41666pt}\pgfsys@lineto{0.0pt}{3.57222pt }\pgfsys@lineto{3.19446pt}{3.57222pt}\pgfsys@lineto{3.19446pt}{-0.41666pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.19444pt\hbox {\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to2.24pt{\vbox to2.79pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt }\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox {\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to1.6pt{\vbox to1.99pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.20833pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.20833pt}\pgfsys@lineto{0.0pt}{1.7861pt} \pgfsys@lineto{1.59723pt}{1.7861pt}\pgfsys@lineto{1.59723pt}{-0.20833pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-1.59721pt\hbox {\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}0}}\left[\sum_{i\neq j }\log\frac{R_{t}(i,j)}{R_{T}(i,j)}\sum_{u=0}^{t}H_{u}^{ij}.\right]$ (20c) $$

In Eq. (20b), we utilized the fact that under $\tau$-leaping, $R^{T\mathrel{\mathchoice{\mkern 2.0mu\leavevmode\leavevmode\hbox{ \leavevmode
\hbox to4.47pt{\vbox to5.58pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower
-0.58333pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{
pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }
\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}
\pgfsys@invoke{ }\nullfont\hbox to0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333
pt}\pgfsys@lineto{0.0pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{5.0011pt}
\pgfsys@lineto{4.47223pt}{-0.58333pt}\pgfsys@closepath\pgfsys@clipnext
\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox
{\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
\leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{
\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{
\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}
{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to
0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt}
\pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt}
\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox
{\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
\leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.13pt{\vbox to3.91pt{
\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.40833pt\hbox to0.0pt{
\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}
{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to
0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.40833pt}\pgfsys@lineto{0.0pt}{3.50076pt
}\pgfsys@lineto{3.13055pt}{3.50076pt}\pgfsys@lineto{3.13055pt}{-0.40833pt}
\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.13055pt\hbox
{\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
\leavevmode\leavevmode\hbox{ \leavevmode\hbox to2.24pt{\vbox to2.79pt{
\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{
\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}
{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to
0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt
}\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt}
\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox
{\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}t\mathrel{
\mathchoice{\mkern 2.0mu\leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt
{\vbox to5.58pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt
\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{
rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }
\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}
\pgfsys@invoke{ }\nullfont\hbox to0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333
pt}\pgfsys@lineto{0.0pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{5.0011pt}
\pgfsys@lineto{4.47223pt}{-0.58333pt}\pgfsys@closepath\pgfsys@clipnext
\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox
{\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
\leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{
\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{
\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}
{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to
0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt}
\pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt}
\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox
{\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
\leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.13pt{\vbox to3.91pt{
\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.40833pt\hbox to0.0pt{
\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}
{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to
0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.40833pt}\pgfsys@lineto{0.0pt}{3.50076pt
}\pgfsys@lineto{3.13055pt}{3.50076pt}\pgfsys@lineto{3.13055pt}{-0.40833pt}
\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.13055pt\hbox
{\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{
\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu
\leavevmode\leavevmode\hbox{ \leavevmode\hbox to2.24pt{\vbox to2.79pt{
\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{
\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}
{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to
0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt
}\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt}
\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}
}{}
{{}{{}}}{{}{}}{}{{}{}}
{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.
0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox
{\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}}
{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }
\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}0}_{u}(i,j)=R_{T}(i,j)$ for $u\in[t,T]$ and $R^{T\mathrel{\mathchoice{\mkern 2.0mu\leavevmode\leavevmode\hbox{ \leavevmode
\hbox to4.47pt{\vbox to5.58pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower
-0.58333pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{
pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }
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{{}{{}}}{{}{}}{}{{}{}}
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#### A.5 Technique 2

Consider the meaning of $\mathbb{E}\left[\sum_{u=0}^{t}H_{u}^{ij}\right]$; it calculates the average probability of a transition from $i$ to $j$ occurring between time 0 0 and $t$. If we knew $\partial_{u}p(x_{u}=j,x_{u-}=i)$, this could be found by $\int_{0}^{t}\partial_{u}p(x_{u}=j,x_{u-}=i)\,du$. However, we do not have access to $\partial_{u}p(x_{u}=j,x_{u-}=i)$.

Fortunately, we do know the conditional transition rate $\partial_{u}p(x_{u}=j\mid x_{u-}=i)=R_{t}(i,j)$. Let’s assume that there are maximally single transition of state in each dimension during the time interval, which is the assumption behind using $\tau$-leaping algorithm for DDMs (Campbell et al., ). Using this, we can rewrite Eq. (20c):

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0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.41666pt}\pgfsys@lineto{0.0pt}{3.57222pt }\pgfsys@lineto{3.19446pt}{3.57222pt}\pgfsys@lineto{3.19446pt}{-0.41666pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.19444pt\hbox {\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to2.24pt{\vbox to2.79pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt }\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox {\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to1.6pt{\vbox to1.99pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.20833pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.20833pt}\pgfsys@lineto{0.0pt}{1.7861pt} \pgfsys@lineto{1.59723pt}{1.7861pt}\pgfsys@lineto{1.59723pt}{-0.20833pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-1.59721pt\hbox {\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}0}}\left[\sum_{i\neq j }\log\frac{R_{t}(i,j)}{R_{T}(i,j)}\sum_{t<u\leq T}H_{u}^{ij}\right]$ $\displaystyle=\mathbb{E}_{\mathbb{Q}_{\mathrm{paths}}^{T\mathrel{\mathchoice{ \mkern 2.0mu\leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.19pt{\vbox to 3.99pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.41666pt\hbox to0.0pt {\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.41666pt}\pgfsys@lineto{0.0pt}{3.57222pt }\pgfsys@lineto{3.19446pt}{3.57222pt}\pgfsys@lineto{3.19446pt}{-0.41666pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.19444pt\hbox {\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.19pt{\vbox to3.99pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.41666pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.41666pt}\pgfsys@lineto{0.0pt}{3.57222pt }\pgfsys@lineto{3.19446pt}{3.57222pt}\pgfsys@lineto{3.19446pt}{-0.41666pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.19444pt\hbox {\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to2.24pt{\vbox to2.79pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt }\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox {\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to1.6pt{\vbox to1.99pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.20833pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.20833pt}\pgfsys@lineto{0.0pt}{1.7861pt} \pgfsys@lineto{1.59723pt}{1.7861pt}\pgfsys@lineto{1.59723pt}{-0.20833pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-1.59721pt\hbox {\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}t}}\left[\mathbb{E}_ {\mathbb{Q}_{\mathrm{paths}}^{t\mathrel{\mathchoice{\mkern 2.0mu\leavevmode \leavevmode\hbox{ \leavevmode\hbox to3.19pt{\vbox to3.99pt{\pgfpicture \makeatletter\hbox{\hskip 0.0pt\lower-0.41666pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.41666pt}\pgfsys@lineto{0.0pt}{3.57222pt }\pgfsys@lineto{3.19446pt}{3.57222pt}\pgfsys@lineto{3.19446pt}{-0.41666pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.19444pt\hbox {\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.19pt{\vbox to3.99pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.41666pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.41666pt}\pgfsys@lineto{0.0pt}{3.57222pt }\pgfsys@lineto{3.19446pt}{3.57222pt}\pgfsys@lineto{3.19446pt}{-0.41666pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.19444pt\hbox {\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to2.24pt{\vbox to2.79pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt }\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox {\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to1.6pt{\vbox to1.99pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.20833pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.20833pt}\pgfsys@lineto{0.0pt}{1.7861pt} \pgfsys@lineto{1.59723pt}{1.7861pt}\pgfsys@lineto{1.59723pt}{-0.20833pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-1.59721pt\hbox {\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}0}}\left[\sum_{i\neq j }\log\frac{R_{t}(i,j)}{R_{T}(i,j)}\sum_{t<u\leq T}H_{u}^{ij}\Bigg{|}X_{t}=i \right]\right]$ (21a) $\displaystyle\approx\mathbb{E}_{\mathbb{Q}_{\mathrm{paths}}^{T\mathrel{ \mathchoice{\mkern 2.0mu\leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.19pt {\vbox to3.99pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.41666pt \hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{ rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ } \pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt} \pgfsys@invoke{ }\nullfont\hbox to0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.41666 pt}\pgfsys@lineto{0.0pt}{3.57222pt}\pgfsys@lineto{3.19446pt}{3.57222pt} \pgfsys@lineto{3.19446pt}{-0.41666pt}\pgfsys@closepath\pgfsys@clipnext \pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}}{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.19444pt\hbox {\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.19pt{\vbox to3.99pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.41666pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.41666pt}\pgfsys@lineto{0.0pt}{3.57222pt }\pgfsys@lineto{3.19446pt}{3.57222pt}\pgfsys@lineto{3.19446pt}{-0.41666pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.19444pt\hbox {\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to2.24pt{\vbox to2.79pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt }\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox {\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to1.6pt{\vbox to1.99pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.20833pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.20833pt}\pgfsys@lineto{0.0pt}{1.7861pt} \pgfsys@lineto{1.59723pt}{1.7861pt}\pgfsys@lineto{1.59723pt}{-0.20833pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-1.59721pt\hbox {\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}t}}\left[\sum_{X_{t} \neq j}\log\frac{R_{t}(X_{t},j)}{R_{T}(X_{t},j)}\times R_{t}(X_{t},j)\Delta t\right]$ (21b) $$

Equation Eq. (21a) applies the Law of Total Expectation, and in Eq. (21b), we utilize the equation:

$$ $\mathbb{E}_{\mathbb{Q}_{\mathrm{paths}}^{t\mathrel{\mathchoice{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.19pt{\vbox to3.99pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.41666pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.41666pt}\pgfsys@lineto{0.0pt}{3.57222pt }\pgfsys@lineto{3.19446pt}{3.57222pt}\pgfsys@lineto{3.19446pt}{-0.41666pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.19444pt\hbox {\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.19pt{\vbox to3.99pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.41666pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.41666pt}\pgfsys@lineto{0.0pt}{3.57222pt }\pgfsys@lineto{3.19446pt}{3.57222pt}\pgfsys@lineto{3.19446pt}{-0.41666pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.19444pt\hbox {\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to2.24pt{\vbox to2.79pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt }\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox {\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to1.6pt{\vbox to1.99pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.20833pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.20833pt}\pgfsys@lineto{0.0pt}{1.7861pt} \pgfsys@lineto{1.59723pt}{1.7861pt}\pgfsys@lineto{1.59723pt}{-0.20833pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-1.59721pt\hbox {\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}0}}\left[\sum_{u=0}^ {t}H_{u}^{ij}\Bigg{|}X_{t}=i\right]\approx R_{t}(i,j)\Delta t,$ $$

where $\Delta t=t-0=t$. The approximation becomes exact when there are only one transition in each dimension during single interval, and this assumption is true for absorb transition matrix.

### Appendix B Algorithm

In this section, we present the main algorithm for Jump your steps.

#### B.1 KLUB computation

Please refer to Algorithm 1.
In the case of $k$-Gillespie, we first sampled the corresponding $t$ using the corresponding $p(t|k)$, and then apply Algorithm 1. Here, $p(t|k)$ determined by predefined noise schedule.

Figure: Algorithm 1 Computation of $\text{KLUB}(\mathbb{Q}^{s\mathrel{\mathchoice{\mkern 2.0mu\leavevmode \leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{\pgfpicture \makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.13pt{\vbox to3.91pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.40833pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.40833pt}\pgfsys@lineto{0.0pt}{3.50076pt }\pgfsys@lineto{3.13055pt}{3.50076pt}\pgfsys@lineto{3.13055pt}{-0.40833pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.13055pt\hbox {\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to2.24pt{\vbox to2.79pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt }\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox {\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}t\mathrel{ \mathchoice{\mkern 2.0mu\leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt {\vbox to5.58pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt \hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{ rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ } \pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt} \pgfsys@invoke{ }\nullfont\hbox to0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333 pt}\pgfsys@lineto{0.0pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{-0.58333pt}\pgfsys@closepath\pgfsys@clipnext \pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}}{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.13pt{\vbox to3.91pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.40833pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.40833pt}\pgfsys@lineto{0.0pt}{3.50076pt }\pgfsys@lineto{3.13055pt}{3.50076pt}\pgfsys@lineto{3.13055pt}{-0.40833pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.13055pt\hbox {\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to2.24pt{\vbox to2.79pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt }\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox {\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}u}\|\mathbb{Q}^{s \mathrel{\mathchoice{\mkern 2.0mu\leavevmode\leavevmode\hbox{ \leavevmode\hbox to 4.47pt{\vbox to5.58pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333 pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor }{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ } \pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt} \pgfsys@invoke{ }\nullfont\hbox to0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333 pt}\pgfsys@lineto{0.0pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{-0.58333pt}\pgfsys@closepath\pgfsys@clipnext \pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}}{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\displaystyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to4.47pt{\vbox to5.58pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.58333pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.58333pt}\pgfsys@lineto{0.0pt}{5.0011pt} \pgfsys@lineto{4.47223pt}{5.0011pt}\pgfsys@lineto{4.47223pt}{-0.58333pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-4.47223pt\hbox {\set@color{$\textstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to3.13pt{\vbox to3.91pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.40833pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.40833pt}\pgfsys@lineto{0.0pt}{3.50076pt }\pgfsys@lineto{3.13055pt}{3.50076pt}\pgfsys@lineto{3.13055pt}{-0.40833pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-3.13055pt\hbox {\set@color{$\scriptstyle\vphantom{+}{\shortrightarrow}$}}}}}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\mkern 2.0mu \leavevmode\leavevmode\hbox{ \leavevmode\hbox to2.24pt{\vbox to2.79pt{ \pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower-0.29166pt\hbox to0.0pt{ \pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0} {0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{-0.29166pt}\pgfsys@lineto{0.0pt}{2.50055pt }\pgfsys@lineto{2.23611pt}{2.50055pt}\pgfsys@lineto{2.23611pt}{-0.29166pt} \pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{} }{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1. 0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip-2.23611pt\hbox {\set@color{$\scriptscriptstyle\vphantom{+}{\shortrightarrow}$}}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope } \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}u})$

#### B.2 Jump your steps

Please refer to Algorithm 2.

Figure: Algorithm 2 Jump Your Steps

### Appendix C Experiment Details

##### Golden Section

The golden section search was stopped if the difference between the newly optimized $t$ and the previous $t$ was smaller than $T/2048$. The maximum number of iterations was set to 32, but usually, the iterations were completed within 8 steps.

##### CountDown

We use SEDD (Lou et al., ) for loss function and the DiT (Peebles & Xie, ) as a model architecture, the noise schedule followed the log-linear scheme proposed in the SEDD paper. KLUB computation was done with $\text{num_samples}=2048$, and one golden section search took approximately 4 seconds.

##### CIFAR10

The pretrained model provided by CTMC (Campbell et al., ) was used.
For CIFAR10, with $\text{num_samples}=1024$, one golden section search took about 30 seconds.

##### Monophonic Music

The pretrained model provided by CTMC (Campbell et al., ) was used. KLUB computation was performed with 2048 samples, and one golden section search took about 20 seconds.

##### Text

The pretrained model provided by SEDD (Lou et al., ) was used.
With $\text{num_samples}=256$, one golden section search took 120 seconds.

### Appendix D Additional Results

#### D.1 Ablation study

Figure: Figure 10: Performance comparisons on Monophonic music.
Refer to caption: x9.png

Figure 10 presents the ablation study on the number of samples for KLUB computation. We compare a diverse set of sample sizes, ranging from 4 to 1024. We optimize the JYS sampling schedule for a CTMC trained on CIFAR-10, which has a Gaussian transition matrix. The results show that there was almost no difference. As shown in Figure 4, this seems to be because the KLUB distribution of each sample shows a similar trend with respect to t, so optimization over t does not vary significantly depending on the number of samples. Based on this, it appears that the JYS sampling schedule optimization can be performed with much fewer resources than the computation used in this paper. For example, we can optimize JYS sampling schedule less than 1 minute if we only use 16 samples.

#### D.2 Qualitative results

In this subsection, we present qualitative comparison between Jump Your Steps and Uniform sampling schedule under various NFEs.

##### CIFAR10

Figure 11 shows generated images with various NFE and sampling schedule. All results are generated using Euler $\tau$-leaping sampler.

Figure: Figure 11: CIFAR10 results.
Refer to caption: x10.png

##### Text

Figure 12, 13, 14, 15 show generated text samples with various NFE and sampling schedule. All results are generated using Euler $\tau$-leaping sampler.

Figure: Figure 12: Generated samples using uniform schedule (NFE = 64).
Refer to caption: x11.png

Figure: Figure 13: Generated samples using Jump Your Steps schedule (NFE = 64).
Refer to caption: x12.png

Figure: Figure 14: Generated samples using uniform schedule (NFE = 256).
Refer to caption: x13.png

Figure: Figure 15: Generated samples using Jump Your Steps schedule (NFE = 256).
Refer to caption: x14.png

## Peer Reviews

confidence: 4
contribution: 2
initial_rating: 5
presentation: 3
questions: - The proposed scheduling strategy only considers $2^K$ numbers of function evaluations (NFE). How can the schedule extend to other values of NFE? 
- In Figure 5, is there any intuition to explain the error increase of the $k$-Gillespie sampler with the JYS schedule when NFE=16? 
- In Figure 10, the comparison between uniform and JYS is not significant for NFE=32, 128, or 256. Please consider providing additional experiments on other image datasets, e.g., toy 2D datasets in Sabour et al. (2024).
soundness: 2
strengths: This paper proposes an approach to improve the sampling quality and efficiency by optimizing the sampling schedule. This paper is well-written and easy to follow. The sampling and acceleration of diffusion models is an interesting topic for the community.
summary: This paper proposes a method, Jump Your Steps, to optimize the discretization schedule of discrete diffusion models, by minimizing the compounding decoding error (CDE). The authors provide a KL-divergence upper bound (KLUB) for different continuous-time Markov chains of discrete diffusion models and propose techniques to efficiently approximate the timesteps minimizing KLUB.
weaknesses: - Theorem 3.1 seems to be standard entropy bounds, and Theorem 3.2 seems to directly follow from Equations 3.2 and 3.4 of Ding & Ning (2021) and Theorem 3.2 of Sabour et al. (2024). The authors should illustrate their technical contributions and the novelty of their theoretical results with a comparison to the previous literature. 
- The authors only compare the JYS schedule with the uniform schedule. Additional experiments on other schedules, e.g. EDM, Linear LogSNR, and Cosine LogSNR, should be presented.

---

confidence: 4
contribution: 2
initial_rating: 3
presentation: 2
questions: N/A
soundness: 2
strengths: The algorithm presented in this submission is straightforward and simple. Experimental results have been provided to demonstrate the feasibility of the proposed algorithm.
summary: This paper introduces Jump Your Steps (JYS), a novel approach that optimizes the allocation of discrete sampling timesteps. It achieves this by minimizing Compounding Decoding Error (CDE) without incurring any additional computational cost. The authors derive an upper bound on CDE and employ techniques to simplify the computation. Based on the upper bound, they propose an algorithm for searching for the optimal sampling schedule, thereby enhancing the sampling quality.
weaknesses: 1. The novelty of the proposed algorithm is rather limited. The authors merely transfer Align Your Steps (AYS), which is originally for image generation, to the framework of SEDD for text generation.

2. The experiments conducted on the CIFAR-10 and Countdown datasets lack practical significance. In the realm of image generation, efficient sampling methods have been well explored. For CIFAR-10, even with 256 NFEs, the proposed approach can only achieve an FID higher than 10. In contrast, some popular sampling methods for image generation, such as DPM-Solver++ (https://arxiv.org/pdf/2211.01095) and DMD (https://arxiv.org/pdf/2311.18828), can attain an FID less than 4 using only 10 or even just 1 sampling step. Additionally, the Countdown dataset is a synthetic one. I suggest that the authors place the results for CIFAR-10 and Countdown in the appendix. Moreover, as I am not familiar with music generation, it appears to me that although JYS can lead to a significant drop in Hellinger distance, the Hellinger distance for uniform sampling timesteps is already small (less than 0.4).

3. It seems to me that the practically meaningful experimental results are mainly showcased in Figure 8 for text generation. Nevertheless, upon examining Figure 8, I cannot observe a significant improvement over the use of uniform sampling timesteps.

4. Some problems in the writing:
- The right arrows used on Line 52 seem strange. Additionally, such right arrows occur multiple times in the main content. It is suggested to use normal notations instead (and I don't think there is a need to write down the whole sequence for timesteps repeatedly).
- The term "KLUB" is first seen in Figure 2 on page 4; however, its explanation is only provided in Theorem 3.1 on page 5.
- There is no necessity to write down the Sampler in Figures 6 and 7.
- In the section on "$k$-Gillespie's Algorithm" (Line 125), '$k$' apparently means updating $k$ tokens in parallel. However, in Appendix B.1 KLUB COMPUTATION, the use of '$p(t|k)$' is presented and it is explained that this is determined by the pretrained noise schedule. As a result, I am not certain whether '$t$' signifies time. If it does, then it appears inconsistent with '$k$'. Please provide a more detailed explanation about $k$-Gillespie's Algorithm.
- The condition $\mathbb{P}_{t_0} = \mathbb{Q}_{t_0}$ in the last sentence of the proof of Theorem 3.1 should be presented as an assumption in the statement of Theorem 3.1.


--------

After rebuttals:

I would like to thank the authors for their detailed responses. However, the authors have only provided Figure 13 (which has been shifted to illustrate the performance gains of JYS compared to those achieved by increasing model size) to address my major concern in W3. Currently, although I have slightly raised my score to 5, my concern about the practical significance and relevance of the proposed approach still persists. As a result, I am still inclined towards rejection.

---

confidence: 4
contribution: 3
initial_rating: 8
presentation: 3
questions: 1. As the empirical validation of the KLUB is a proxy for CDE, would it be possible to directly show that minimizing KLUB actually reduces this error?
2. Does the hierarchical breakdown strategy guarantee convergence to a globally optimal schedule?
3. The paper says “After K iterations, this hierarchical strategy yields a sampling schedule with 2^K NFEs, optimizing the schedule as the number of steps increases.” in lines 289-29. So, there is an exponential factor appearing in the number of function estimations. I wonder if it affects the overall complexity and method’s applicability, especially in resource-constrained settings.
4. I also wonder if the computational gain from this approach leads to higher memory costs. Is it this case from your observations?
5. Is there any intuition on how the method's error bounds scale with data size, or it doesn’t matter?
6. Is there any insight into the applicability of the proposed method to DDMs without fast sampling algorithms?
7. Do you think the hierarchical breakdown strategy converges to globally optimal schedules?

Small suggestions regarding the manuscript:
- There is no need to define DDMs (lines 12, 30, 94, for example) or CDE (line 156) a few times through the text if they were defined before.
- In Eq. 1 and this section, I’d suggest explicitly defining $x$ (data) and $y$ (corrupted data) variables.
- Line 133: Should it be “.” instead of “:”? 
- Maybe notations in line 148 can be combined with Section 2.1 or moved there? I’d suggest the authors go through their manuscript to make sure all notations are introduced and consistent.
NFE is not defined in line 290 or earlier. Also, it is useful to explain what it means (it is the number of calls to the score NN, right?).
- Maybe it’s worth moving Section 5 to the Intro section or a separate section at the beginning of the manuscript? Also, mentioning papers [1, 2] might be helpful.
- Adding equation numbers to Algorithm 1(line 7) would be helpful.
soundness: 4
strengths: The main idea of this paper is novel: the authors suggest reframing DDM sampling schedule optimization as a CDE minimization problem, introducing a Kullback-Leibler divergence upper bound (KLUB) metric to make the optimization tractable for improving sample quality in fast DDM sampling methods. The theoretical development seems sound, establishing connections between CDE minimization and path-space probability measures. This work is particularly interesting because it requires no architectural modifications or additional inference costs, unlike existing methods. As for the empirical validation, the paper includes an evaluation across diverse domains showing consistent improvements in the sample quality when using the same number of function evaluations (NFEs) as a baseline approach.
summary: This paper introduces Jump Your Steps (JYS), a method for discrete diffusion models (DDMs) that enhances generation quality by finding optimal sampling schedules when using efficient parallel sampling methods. Parallel sampling methods such as τ-leaping can speed up DDM inference, but they introduce a Compounding Decoding Error (CDE) that demotes sample quality. The authors address this by deriving a practical upper bound on CDE and developing an efficient schedule optimization algorithm that minimizes this bound without additional computational overhead. They perform experiments on image, music, and text generation tasks to show that the JYS approach improves sampling quality across different transition kernels using fast sampling methods like τ-leaping or k-Gillespie.
weaknesses: 1. The empirical analysis can be improved: 
 - While improvements in the generation quality in terms of FID or perplexity scores are shown, comparisons against other quality-improvement methods (e.g., predictor-corrector approaches mentioned in Section 1), and an explicit study of their computational costs are missing. The paper should either include such comparisons or justify why they aren't necessary.
 - The trade-off between schedule optimization time (or its memory requirements/end-to-end generation time) and sample quality gains isn't studied, making it difficult to assess the method's practical utility.
- No error bars or standard deviations are provided for the quantitative metrics. This is important since generative models can have high variance in their outputs and metrics. An elaboration of the significance of the reported results would be appreciated. 
-- The comparison is limited to a uniform scheduling baseline. Would it be possible to compare against an adaptive step size integrator method (Runge–Kutta method modifications if they are applicable to the discrete case) or other works like a recent one [5]? It’d be great to include this or explain why such comparisons aren't applicable or necessary.
2. The paper doesn't clarify how the diffusion model formulation (Eq. 1) relates to other DDM variants from [2, 3, 4]. Such discussion would be appreciated.
3. The single-state transition assumption, while mathematically convenient, limits the method's applicability: it can lead to suboptimal schedules for fast-changing processes. It’d be great to see ideas on possible extensions to handle multiple transitions while maintaining computational tractability.
4. Some technical details are difficult to follow or verify. For example, the authors introduce two techniques for KLUB computation in Section 3.5 but don't explain when each is preferable. It looks like both techniques are combined in Algorithm 1. No discussion of whether the hierarchical breakdown strategy converges to globally optimal schedules. The clarity of the paper could also be improved. There are suggestions on this in the Questions section.
5. No code is provided for reproducibility. 


[1] Hoogeboom, Emiel, et al. "Argmax flows and multinomial diffusion: Learning categorical distributions." Advances in Neural Information Processing Systems 34 (2021): 12454-12465.

[2] Santos, Javier E., et al. "Blackout diffusion: generative diffusion models in discrete-state spaces." International Conference on Machine Learning. PMLR, 2023.

[3] Austin, Jacob, et al. "Structured denoising diffusion models in discrete state-spaces." Advances in Neural Information Processing Systems 34 (2021): 17981-17993.

[4] Hoogeboom, Emiel, et al. "Autoregressive diffusion models." arXiv preprint arXiv:2110.02037 (2021).

[5] Chen, Yuzhu, et al. "Adaptive Time-Stepping Schedules for Diffusion Models." The 40th Conference on Uncertainty in Artificial Intelligence.

---

confidence: 4
contribution: 2
initial_rating: 6
presentation: 3
questions: 1) line 189 saying "motivated by this observation" is not correct/precise I think, given that you list a bunch of factors at work and you just focus on one? the only motivating point is the dependence on sampling methods, and not the others.

2) line 202 "if we ignore the accumulated error from the previous steps": why should we ignore it? can you formulate this as an assumption somewhere?

I think that the paper's theory part might have very non-sharp estimates in case this error can't be ignored, can you convince me otherwise?

3) can you comment on when is the inequality in (7) sharp? I think this has to do with question 2.

4) line 216-217 "this theorem suggests" -- I think that it'd be accurate to say "based on the case that inequality (7) is sharp, we are tempted to hypothesize" and to mention that in case accumulated error from previous steps is negligible, then this heuristic becomes stronger.

Note that in applications in which the gap in (7) is large, then theorem 3.1 gives a lax and uninformative upper bound and this heuristic has no reason for holding.

5) about Theorem 3.2, when is the inequality in (8) sharp? Upper bounds like KLUB are useful to the extent to which they are close to sharp. So this limits the range of applicability of (8), and therefore it would be good to have a description of cases in which this is sharp.

6) about the hierarchical time breakdown (described in paragraph 3.4 and summarized in Figure 3) : this is one possible time breakdown strategy. Can you comment on alternative strategies, or on how this one is better than, or equivalent to, other strategies? 

7) line 298-299 what do you mean by "computational complexities"? This is vague and easy to misinterpret for a reader. I think you mean the thing mentioned in lines 350-360? If so, I suggest moving those lines up here, so that the reader has a concrete thing in mind since the beginning.

8) about the formula/equation at lines 308-309: what does the "approximately equal" sign actually mean? and why or under what hypotheses is this approximation valid? You mention something in the preceding paragraph, but this is not giving a clear cut answer to what approximation you have in mind, and on what are the underlying assumptions behind it.

9) about figure 9, I am puzzled/perplexed by the cases (a) and (c): 
In (a), why do you think that the JYS schedule has so dense subdivision in the first interval? I don't see anything in the "ground truth steps" graph that would indicate that this is optimal. 
In (c), why is the JYS deciding to just not include any timesteps in the first interval? In there, I think that the Ground Truth Steps part has a very dense time dependence, at least that's what I see in the figure.. so how do you justify this discrepancy?

To summarize, both points (a) and (c) show that JYS has high discrepancy in its "optimized" timestep choice, compared to "ground truth samples", so it would be good to understand what this means, how do we interpret this.
soundness: 3
strengths: The direction of this research is new and interesting.
summary: The paper describes a way to optimize sampling for discrete diffusion models. The idea is to select a good subset of optimal sample times based on an optimization criterion. Some theory and some experiments are given.
weaknesses: Some of the theory parts confounds heuristic and rigorous justification (by ignoring some long-time dependencies without actually emphasizing that this is a working simplification). Thus, some assumptions and/or limitations are not highlighted, which is making the paper look weaker. ----[update] this was addressed in the discussion, with reference to appendices

Also, I am not completely convinced with the interpretation of some of the experiments, but this may be just me. --- [update] the authors clarified and promised to amend some figure

Meta-review: This work addresses the challenge of slow sampling speeds in discrete diffusion models (DDMs) by introducing Jump Your Steps (JYS), a novel method to optimize discrete sampling timesteps. JYS minimizes Compounding Decoding Error (CDE), a key issue in parallel sampling methods like τ-leaping, which degrade sample quality. The authors derive a practical upper bound on CDE and propose an efficient algorithm to determine an optimal sampling schedule, all without additional computational cost. Experiments across image, music, and text generation demonstrate that JYS significantly improves sampling quality. Following the author response and author-reviewer discussions, the reviews are still mixed. Considering the substantial theoretical contributions, which outweigh the relatively weaker empirical experiments, I recommend to accept.

In the final version, I encourage the authors to include a discussion on DNDM, proposed by [Chen et al., 2024]. This closely related work shares a similar concept of utilizing predetermined transition times (i.e., allocating discrete sampling timesteps) to accelerate the sampling process in discrete diffusion models.

[Chen et al. 2024] Fast Sampling via Discrete Non-Markov Diffusion Models with Predetermined Transition Time

Decision: Accept (Poster)