SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Prateek Jaiswal
jaiswalp@stat.tamu.edu
Harsha Honnappa
West Lafayette, IN, 47906
honnappa@purdue.edu
Vinayak A. Rao
varao@purdue.edu
We study data-driven decision-making problems in the Bayesian framework, where the expectation in the Bayes risk is replaced by a risk-sensitive entropic risk measure with respect to the posterior distribution. We focus on problems where calculating the posterior distribution is intractable, a typical situation in modern applications with large datasets and complex data generating models. We leverage a dual representation of the entropic risk measure to introduce a novel risk-sensitive variational Bayesian (RSVB) framework for jointly computing a risk-sensitive posterior approximation and the corresponding decision rule. Our general framework includes loss-calibrated VB [16] as a special case. We also study the impact of these computational approximations on the predictive performance of the inferred decision rules. We compute the convergence rates of the RSVB approximate posterior and the corresponding optimal value. We illustrate our theoretical findings in parametric and nonparametric settings with the help of three examples.
This paper focuses on risk-sensitive Bayesian decision-making, considering objective functions of the form
$$
\min  _ {a \in \mathcal {A}} \varrho_ {\Pi_ {n}} ^ {\gamma} (R (a, \theta)) := \gamma^ {- 1} \log \mathbb {E} _ {\Pi_ {n}} [ \exp (\gamma R (a, \theta)) ]. \tag {SO}
$$
Here  $\mathcal{A}$  is the decision/action space,  $\theta$  is a random model parameter lying in an arbitrary measurable space  $(\Theta, \mathcal{T})$ , and  $R(a, \theta): \mathcal{A} \times \Theta \mapsto \mathbb{R}$  is a problem-specific model risk function. For any  $B \subseteq \Theta$ ,  $\Pi_n(B) := \Pi(B|\tilde{X}_n)$  is the Bayesian posterior distribution over the parameters given observations  $\tilde{X}_n$  from the true model  $P_{\theta_0}^n (\equiv P_0^n)$  with parameter  $\theta_0 \in \Theta$ . The scalar  $\gamma \in \mathbb{R}$  is user-specified and characterizes the sensitivity of the decision-maker (DM) to the distribution  $\Pi_n$ . Recall that the posterior distribution is obtained by updating a prior probability distribution  $\Pi(\cdot)$ , capturing subjective beliefs of the decision maker over  $\Theta$ , according to the Bayes rule
$$
d \Pi (\theta | \tilde {X} _ {n}) \propto d \Pi (\theta) d P _ {\theta} ^ {n} (\tilde {X} _ {n}), \tag {1}
$$
where  $dP_{\theta}^{n}(\tilde{X}_{n})$  is the likelihood of observing  $\tilde{X}_n$ . We denote the corresponding densities (if they exist) for the model, prior, and posterior as  $p_{\theta}^{n}(\cdot),\pi (\cdot)$ , and  $\pi (\cdot |\tilde{X}_n)$  respectively.
The functional  $\varrho^{\gamma}$  in (SO) is also known as the entropic risk measure, and models a range of risk-averse or risk-seeking behaviors in a succinct manner through the parameter  $\gamma$ . Consider only strictly positive  $\gamma$ , and observe that  $\lim_{\gamma \downarrow 0} \frac{1}{\gamma} \log \mathbb{E}_{\Pi_n}[\exp (\gamma R(a,\theta))] = \mathbb{E}_{\Pi_n}(R(a,\theta))$ ; that is, there is no sensitivity to potential risks due to large tail effects and the decision-maker is risk neutral.
On the other hand,  $\lim_{\gamma \to +\infty}\varrho_{\Pi_n}^\gamma (R(a,\theta)) = \operatorname *{ess}\sup_{\Pi_n}(R(a,\theta))$  , the essential supremum of the model risk  $R(a,\theta)$  . In other words, a decision maker is completely risk averse and anticipates the worst possible realization  $(\Pi_{n}$  -almost surely). Observe that (SO) strictly generalizes the standard Bayesian decision-theoretic formulation of a decision-making problem, where the goal is to solve  $\min_{a\in \mathcal{A}}\mathbb{E}_{\Pi_n}[R(a,\theta)]$
The risk-sensitive formulation (SO) is very general and can be used to model a wide variety of decision-making problems in machine learning [16, 15], operations research/management science [24, 7, 20], simulation optimization [6, 32], and finance [17, 1, 4]. However, solving (SO) to compute an optimal decision over  $\mathcal{A}$  is challenging. The difficulty mainly stems from the fact that, with the exception of conjugate models, the posterior distribution in (1) is an intractable quantity. Canonically, posterior intractability is addressed using either a sampling- or optimization-based approach. Sampling-based approaches, such as Markov chain Monte Carlo (MCMC), offer a tractable way to compute the integrals and theoretical guarantees of exact inference in the large computational budget limit.
Optimization-based methods such as variational Bayes (VB) or variational inference (VI) have emerged as a popular alternative [3]. The VB approximation of the true posterior is a tractable distribution chosen from a 'simplex' family of distributions known as a variational family by minimizing the discrepancy between the true posterior and members of that family. Kullback-Liebler (KL) divergence is the most often used measure of the approximation discrepancy, although other divergences (such as the  $\alpha$ -Renyi divergence [19, 28, 14]) have been used. The minimizing member, termed the VB approximate posterior, can be used as a proxy for the true posterior. Empirical studies have shown that VB methods are computationally faster and far more scalable to higher-dimensional problems and large datasets. Theoretical guarantees, such as large sample statistical inference, have been a topic of recent interest in the theoretical statistics community, with asymptotic properties such as convergence rate and asymptotic normality of the VB approximate posterior recently established in [34, 21] and [31, 14] respectively.
Our ultimate goal is not to merely approximate the posterior distribution but also to make decisions when that posterior is intractable. A naïve approach would be to plug in the VB approximation in place of the true posterior in (SO) and compute the optimal decision. However, it has been noted in [16] that such a naïve loss-unaware approach can be suboptimal. In particular, [16] demonstrated, through an example, that a naïve posterior approximation only captures the most dominant mode of the true posterior, which may not be relevant from a decision-making perspective. Consequently, they proposed a loss-calibrated variational Bayesian (LCVB) algorithm for solving Bayesian decision-making problems where the underlying risk function is discrete. [15] extended their approach to continuous risk functions. Despite these algorithmic advances in developing decision-centric VB methods, their statistical properties, such as asymptotic consistency and convergence rates of the loss-aware posterior approximation and the associated decision rule are not well understood. With an aim to address these gaps, we summarize our contribution in this paper below:
1. In Section 2, we introduce a minimax optimization framework titled 'risk-sensitive variational Bayes' (RSVB), extracted from the dual representation of (SO) using the so-called Donsker-Varadhan variational free-energy principle [9]. The decision-maker computes a risk-sensitive approximation to the true posterior (termed as RSVB posterior) and the decision rule simultaneously by solving a minimax optimization problem. Moreover, we recover the LCVB approach [16] as a special case of RSVB (with  $\gamma = 1$ ).
2. In Section 3, we identify verifiable regularity conditions on the prior, likelihood model and the risk function under which the RSVB posterior enjoys the same rate of convergence as the true posterior to a Dirac delta distribution concentrated at the true model parameter  $\theta_0$ , as the sample size increases. Using this result, we also prove the rate of convergence of the RSVB decision rule when the decision space  $\mathcal{A}$  is compact. Our theoretical results also imply the asymptotic properties of the LCVB posterior and the associated decision rule.
3. In Section 4, we demonstrate our theoretical results with the help of three widely studied decision-making problems, including Gaussian process classification and a newsvendor problem. For each example, we show that the rate of convergence of the respective RSVB approximate posterior matches that of the corresponding true posterior.
4. In Section 4, we also present some simulation results with the single product (1-d) newsvendor problem, which is summarized here in Figure 1. The figures demonstrate the effect of changing  $\gamma$  on the optimality gap in values (see Definition 2.1) and the variance of the
RSVB posterior for a given  $n$ . In particular, we observe that for smaller  $n$ , increasing  $\gamma$  (after a certain value) results in a significantly more risk-averse decision; however, the effect of increasing  $\gamma$  on risk-averse decision-making reduces as  $n$  increases. This observation demonstrates the fact that when there is enough certainty in the parameter estimation, the need to be risk-sensitive to parametric uncertainty diminishes. The Figure 1 also compares the optimality gap in values evaluated at RSVB decision rules and variance of the RSVB posterior for various values of  $\gamma$  against the naïve VB approach and against the true posterior when the conjugate prior is used.
(a) Optimality gap in Values
(b) Variance
Figure 1: (a) Optimality gap in values and (b) variance of the RSVB posterior (mean over 100 sample paths) against the number of samples  $(n)$  for various values of  $\gamma$ .
2 Risk-Sensitive Variational Bayes
Our approach exploits the dual representation of the log-exponential risk measure in (SO), which is convex (or extended coherent) risk measure [23, 10]. From the Donsker-Varadhan variational free energy principle [9] observe that,
$$
\varrho_ {\Pi_ {n}} ^ {\gamma} (a) = \max  _ {Q \in \mathcal {M}} \left\{\mathbb {E} _ {Q} [ R (a, \theta) ] - \gamma^ {- 1} \mathrm {K L} (Q | | \Pi_ {n}) \right\} \quad \gamma > 0, \tag {DV}
$$
where  $\mathcal{M}$  is the set of all distribution functions that are absolutely continuous with respect to the posterior distribution  $\Pi_{n}$  and 'KL' represents the Kullback-Leibler divergence. Formally, for any two distributions  $P$  and  $Q$  defined on measurable space  $(\Theta ,\mathcal{T})$ , the KL divergence is defined as  $\mathrm{KL}(Q\| P) = \int_{\Theta}dQ(\theta)\log \frac{dQ(\theta)}{dP(\theta)}$ , when measure  $Q$  is absolutely continuous with respect to  $P$  and  $\infty$  otherwise. Notice that this dual formulation exposes the reason we choose to use the log-exponential risk - the 'free-energy' objective on the RHS provides a combined assessment of the risk associated with model estimation (the KL divergence  $\mathrm{KL}(Q\| \Pi_n)$ ) and the decision risk under the estimated posterior  $Q$  ( $\mathbb{E}_Q[R(a,\theta)]$ ).
As stated above, the reformulation presented in (DV) offers no computational gains. However, restricting ourselves to an appropriately chosen subset  $\mathcal{Q} \subset \mathcal{M}$ , that consists of distributions where the integral  $\mathbb{E}_q[R(a,\theta)]$  and  $\mathrm{KL}(Q||\Pi_n)$  can be tractably computed and optimized, we immediately obtain a risk-sensitive variational Bayesian (RSVB) formulation from (DV):
$$
\gamma^ {- 1} \log \mathbb {E} _ {\Pi_ {n}} \left[ e ^ {\gamma R (a, \theta)} \right] \geq \max  _ {Q \in \mathcal {Q}} \left\{\mathbb {E} _ {Q} [ R (a, \theta) ] - \gamma^ {- 1} \mathrm {K L} (Q | | \Pi_ {n}) \right\} =: \mathcal {F} (a; Q (\cdot), \tilde {X} _ {n}, \gamma). \quad (\mathrm {R S V B})
$$
RSVB is our framework for data-driven risk-sensitive decision-making. The family of distributions  $\mathcal{Q}$  is popularly known as the variational family. Our analysis in Section 3.1 reveals general guidelines on how to choose  $\mathcal{Q}$  that ensures a small optimality gap (defined below) with high probability.
With an appropriate choice of  $\mathcal{Q}$ , the optimization on the RHS can yield a good approximation to the log-exponential risk measurement on the left hand side (LHS). For brevity, for a given  $a\in \mathcal{A}$  we define the RSVB approximation to the true posterior  $\Pi (\theta |\tilde{X}_n)$  as
$$
Q _ {a, \gamma} ^ {*} (\theta | \tilde {X} _ {n}) := \operatorname {a r g m a x} \{Q \in \mathcal {Q}: \mathcal {F} (a; Q (\cdot), \tilde {X} _ {n}, \gamma) \}
$$
and the RSVB optimal decision as
$$
\mathsf {a} _ {\mathbb {R S}} ^ {*} := \operatorname * {a r g m i n} _ {a \in \mathcal {A}} \mathcal {F} (a; Q _ {a, \gamma} ^ {*} (\theta | \tilde {X} _ {n}), \tilde {X} _ {n}, \gamma) = \operatorname * {a r g m i n} _ {a \in \mathcal {A}} \max  _ {Q \in \mathcal {Q}} \mathcal {F} (a; Q (\cdot), \tilde {X} _ {n}, \gamma).
$$
Observe that  $Q_{a,\gamma}^{*}(\theta |\tilde{X}_{n})$  and  $\mathfrak{a}_{\mathbb{R}\mathbb{S}}^*$  are random quantities, conditional on the data  $\tilde{X}_n$ .
Examples of  $\mathcal{Q}$  include the family of Gaussian distributions, delta functions, or the family of factorized 'mean-field' distributions that discard correlations between components of  $\theta$ . The choice of  $\mathcal{Q}$  is decisive in determining the performance of the algorithm. Part of the analysis in this paper is to articulate sufficient conditions on  $\mathcal{Q}$  that ensure small optimality gap (defined below) for the optimal decision,  $\mathsf{a}_{\mathtt{RS}}^*$ . This establishes the statistical "goodness" of the procedure as number of samples increase. In this paper, we analyze the efficacy of the decision rules obtained using the RSVB approximation, by providing high-probability bounds on the optimality gap. We define the optimality gap for any  $\mathsf{a} \in \mathcal{A}$  with value  $V = R(\mathsf{a}, \theta_0)$  as,
Definition 2.1 (Optimality Gap). Let  $V_0^* \coloneqq \min_{a \in \mathcal{A}} R(a, \theta_0)$  be the optimal value for the true model parameter  $\theta_0$ . Then, the optimality gap in the value is the difference  $V - V_0^*$ .
A similar performance measure was used in [15], to measure the effectiveness of loss-calibrated VB (LCVB) approach, which can be obtained by setting  $\gamma = 1$ . We provide definitions of some standard terminologies that we use in the subsequent sections, such as covering number, test functions,  $\Gamma$ -convergence, and primal feasibility in the Appendix A for ready reference. In the following section, we lay down important assumptions used throughout the paper to establish our theoretical results.
In order to bound the optimality gap, we require some control over how quickly the posterior distribution concentrates at the true parameter  $\theta_0$ . Our next assumption in terms of a verifiable test condition on the model (sub-)space is one of the conditions required to quantify this rate. Let  $L_{n}:\Theta \times \Theta \mapsto [0,\infty)$  be a distance metric on the model space.
Assumption 2.1 (Model identifiability). Fix  $n \geq 1$ . Then, for any  $\epsilon > \epsilon_{n}$  such that  $\epsilon_{n} \to 0$  as  $n \to \infty$  and  $n\epsilon_{n}^{2} \geq 1$ , there exists a measurable sequence of test functions  $\phi_{n,\epsilon}: \tilde{X}_{n} \mapsto [0,1]$  and sieve set  $\Theta_{n}(\epsilon) \subseteq \Theta$  such that (i)  $\mathbb{E}_{P_0^n}[\phi_{n,\epsilon}] \leq C_0 \exp(-C n\epsilon^2)$ , and (ii)  $\sup_{\{\theta \in \Theta_n(\epsilon): L_n(\theta, \theta_0) \geq C_1 n\epsilon^2\}} \mathbb{E}_{P_\theta^n}[1 - \phi_{n,\epsilon}] \leq \exp(-C n\epsilon^2)$ .
Observe that Assumption 2.1(i) quantifies the rate at which a Type I error diminishes with the sample size, while the condition in Assumption 2.1(ii) quantifies that of a Type II error. Notice that both of these are stated through test functions; indeed, what is required are consistent test functions. [11, Theorem 7.1] (stated in Appendix as Lemma C.7 for completeness) roughly implies that an appropriately bounded model subspace  $\{P_{\theta}^{n}, \theta \in \Theta\}$  guarantees the existence of consistent test functions, to test the null hypothesis that the true parameter is  $\theta_0$  against an alternate hypothesis – the alternate being defined using the distance function  $L_{n}$ . Subsequently, we will use a specific distance function to obtain finite sample bounds for the optimality gap in decisions and values. Note that in some cases, it is also possible to construct consistent test functions directly without recourse to Lemma C.7. We demonstrate this in Section 4.1 below. Next, we assume a condition on the prior distribution that ensures that it provides sufficient mass to the sieve set  $\Theta_n(\epsilon) \subseteq \Theta$ , as defined above in Assumption 2.1.
Assumption 2.2. Fix  $n \geq 1$ . Then, for any  $\epsilon > \epsilon_n$  such that  $\epsilon_n \to 0$  as  $n \to \infty$  and  $n\epsilon_n^2 \geq 1$ , the prior distribution satisfies  $\Pi(\Theta_n^c(\epsilon)) \leq \exp(-Cn\epsilon^2)$ .
Notice that Assumption 2.2 is trivially satisfied if  $\Theta_n(\epsilon) = \Theta$ . The next assumption ensures that the prior distribution places sufficient mass around a neighborhood of the distribution  $P_0^n$ .
Assumption 2.3 (Prior thickness). Fix  $n \geq 1$  and a constant  $\lambda > 0$ . Let  $A_{n} \coloneqq \left\{\theta \in \Theta : D_{1 + \lambda}(P_{0}^{n}\| P_{\theta}^{n}) \leq C_{3}n\epsilon_{n}^{2}\right\}$ , where  $D_{1 + \lambda}(P_0^n \| P_\theta^n) \coloneqq \frac{1}{\lambda}\log \int \left(\frac{dP_0^n}{dP_\theta^n}\right)^{\lambda}dP_0^n$  is the Rényi divergence between  $P_0^n$  and  $P_{\theta}^{n}$ , assuming  $P_0^n$  is absolutely continuous with respect to  $P_{\theta}^{n}$ . The prior distribution satisfies  $\Pi (A_n) \geq \exp (-nC_2\epsilon_n^2)$ .
This assumption guarantees that the prior distribution covers the neighborhood with positive mass. The above three assumptions are adopted from [11] and has also been used in [34] to prove convergence
rates of variational posteriors. Interested readers may refer to [11] and [34] to read more about the above assumptions.
It is apparent by the first term in (RSVB) that in addition to Assumption 2.1, 2.2, and 2.3, we also require regularity conditions on the risk function  $R(a,\cdot)$ . The next assumption restricts the prior distribution with respect to  $R(a,\theta)$ .
Assumption 2.4. Fix  $n \geq 1$  and  $\gamma > 0$ . For any  $\epsilon > \epsilon_n$ ,  $a \in \mathcal{A}$ ,  $\mathbb{E}_{\Pi}[\mathbb{1}_{\{\gamma R(a,\theta) > C_4(\gamma)n\epsilon^2\}}e^{\gamma R(a,\theta)}] \leq \exp(-C_5(\gamma)n\epsilon^2)$ , where  $C_4(\gamma)$  and  $C_5(\gamma)$  are scalar positive functions of  $\gamma$ .
Note that the set  $\{\gamma R(a,\theta) > C_4(\gamma)n\epsilon^2\}$  represents the subset of the model space where the risk  $R(a,\theta)$  (for a fixed decision  $a$ ) is large, and the prior is assumed to place sufficiently small mass over such sets. Moreover, using the Cauchy-Schwarz inequality observe that  $\mathbb{E}_{\Pi}[\mathbb{1}_{\{\gamma R(a,\theta) > C_4(\gamma)n\epsilon^2\}}e^{\gamma R(a,\theta)}]\leq \left(\mathbb{E}_{\Pi}[\mathbb{1}_{\{\gamma R(a,\theta) > C_4(\gamma)n\epsilon^2\}}]\right)^{1 / 2}\left(\mathbb{E}_{\Pi}[e^{2\gamma R(a,\theta)}]\right)^{1 / 2}\leq$ $e^{-C_4(\gamma)n\epsilon^2}\mathbb{E}_{\Pi}[e^{2\gamma R(a,\theta)}]$ , which implies that if the risk function is bounded in  $(a,\theta)$ , then the assumption is trivially satisfied. Finally, we also require the following condition lower bounding the risk function  $R$ .
Assumption 2.5.  $R(a,\theta)$  is assumed to satisfy  $W\coloneqq \inf_{\theta \in \Theta}\inf_{a\in \mathcal{A}}e^{R(a,\theta)} > 0$
Note that any risk function which is bounded from below in both the arguments satisfies this condition. In the next section, we establish high-probability bounds on the optimality gap in values and decision rules computed using RSVB approach for sufficiently large  $n$ .
3 Asymptotic Analysis of the Optimality Gaps
Our first result, establishes an upper bound on the expected deviation from the true model  $P_0^n$ , measured using distance function  $L_{n}(\cdot ,\theta_{0})$ , under the RSVB approximate posterior. We also note that the following result generalizes Theorem 2.1 of [34], which is exclusively for the case when  $\gamma \rightarrow 0^{+}$ . Our proof techniques follows that of Theorem 2.1 in [34].
Theorem 3.1. Fix  $a' \in \mathcal{A}$  and  $\gamma > 0$ . For any  $L_{n}(\theta, \theta_{0}) \geq 0$ , under Assumptions 2.1, 2.2, 2.3, 2.4, and 2.5, and for  $\min(C, C_4(\gamma) + C_5(\gamma)) > C_2 + C_3 + C_4(\gamma) + 2$  and  $\eta_{n}^{R}(\gamma) := \frac{1}{n} \inf_{Q \in \mathcal{Q}} \mathbb{E}_{P_{0}^{n}}\left[\mathrm{KL}(Q(\theta) \| \Pi(\theta | \tilde{X}_{n})) - \gamma \inf_{a \in \mathcal{A}} \mathbb{E}_{Q}[R(a, \theta)]\right]$ , for sufficiently large  $n$  the RSVB approximator of the true posterior  $Q_{a',\gamma}^{*}(\theta | \tilde{X}_{n})$  satisfies,
$$
\mathbb {E} _ {P _ {0} ^ {n}} \left[ \int_ {\Theta} L _ {n} \left(\theta , \theta_ {0}\right) d Q _ {a ^ {\prime}, \gamma} ^ {*} (\theta | \tilde {X} _ {n}) \right] \leq n \left(M (\gamma) \epsilon_ {n} ^ {2} + M \eta_ {n} ^ {R} (\gamma)\right), \tag {2}
$$
for a positive number  $M(\gamma) = 2\left(C_1 + MC_4(\gamma)\right)$ , where  $M = \frac{2C_1}{\min(C, \lambda, 1)}$ .
Proof sketch. For brevity we denote the likelihood ratio as  $\mathcal{L}\mathcal{R}_n(\theta ,\theta_0) = \frac{p(\tilde{X}_n|\theta)}{p(\tilde{X}_n|\theta_0)}$ . We prove this result using a series of lemma. The first Lemma C.1 separates the term  $\eta_n^R (\gamma)$ , which is later analyzed using Assumption 3.1 in Section 3.1. Lemma C.1 establish that
$$
\begin{array}{l} \zeta \mathbb {E} _ {P _ {0} ^ {n}} \left[ \int_ {\Theta} L _ {n} (\theta , \theta_ {0}) d Q _ {a ^ {\prime}, \gamma} ^ {*} (\theta | \tilde {X} _ {n}) \right] \leq \log \mathbb {E} _ {P _ {0} ^ {n}} \left[ \int_ {\Theta} e ^ {\zeta L _ {n} (\theta , \theta_ {0})} \frac {e ^ {\gamma R (a ^ {\prime} , \theta)} \mathcal {L R} _ {n} (\theta , \theta_ {0}) d \Pi (\theta)}{\int_ {\Theta} e ^ {\gamma R (a ^ {\prime} , \theta)} \mathcal {L R} _ {n} (\theta , \theta_ {0}) d \Pi (\theta)} \right] \\ + \log \mathbb {E} _ {P _ {0} ^ {n}} \left[ \int_ {\Theta} e ^ {\gamma R \left(a ^ {\prime}, \theta\right)} \frac {\mathcal {L} \mathcal {R} _ {n} \left(\theta , \theta_ {0}\right) d \Pi (\theta)}{\int_ {\Theta} \mathcal {L} \mathcal {R} _ {n} \left(\theta , \theta_ {0}\right) d \Pi (\theta)} \right] + n \eta_ {n} ^ {R} (\gamma). \tag {3} \\ \end{array}
$$
The proof of Lemma C.1 uses simple arguments that follow easily from Jensen's inequality and the definition of the posterior distribution.
To analyze the first term in the above display, we define the set  $K_{n} \coloneqq \{\theta \in \Theta : L_{n}(\theta, \theta_{0}) > C_{1} n\epsilon^{2}\}$ , with an aim to control the exponential moment of  $L_{n}(\theta, \theta_{0})$  by characterizing its tails. Also, notice that set  $K_{n}$  is the set of alternate hypotheses as defined in Assumption 2.1 through Lemma C.2. For brevity, define  $\Pi_{R}(K_{n}|\tilde{X}_{n}) \coloneqq \frac{\int_{K_{n}} e^{\gamma R(a', \theta)} \mathcal{L}\mathcal{R}_{n}(\theta, \theta_{0}) d\Pi(\theta)}{\int_{\Theta} e^{\gamma R(a', \theta)} \mathcal{L}\mathcal{R}_{n}(\theta, \theta_{0}) d\Pi(\theta)}$ . Next, we divide the expected calibrated posterior probability of the set  $K_{n}$  as follows:
$$
\mathbb {E} _ {P _ {0} ^ {n}} \left[ \Pi_ {R} \left(K _ {n} \mid \tilde {X} _ {n}\right) \right] \leq \mathbb {E} _ {P _ {0} ^ {n}} \phi_ {n, \epsilon} + \mathbb {E} _ {P _ {0} ^ {n}} \left[ \mathbb {1} _ {B _ {n} ^ {C}} \right] + \mathbb {E} _ {P _ {0} ^ {n}} \left[ \left(1 - \phi_ {n, \epsilon}\right) \mathbb {1} _ {B _ {n}} \Pi_ {R} \left(K _ {n} \mid \tilde {X} _ {n}\right) \right], \tag {4}
$$