SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Theo Gruner  $^{1,2}$  Boris Belousov  $^{3}$  Fabio Muratore  $^{4}$  Daniel Palenicek  $^{1,2}$  Jan Peters  $^{1,2,3,5}$
$^{1}$  Intelligent Autonomous Systems Group, Technical University of Darmstadt  
 $^{2}$  hessian.AI 3 German Research Center for AI (DFKI)  
 $^{4}$  Bosch Center for Artificial Intelligence 5 Centre for Cognitive Science theo_sunao.gruner@tu-darmstadt.de
Simulation-Based Inference (SBI) is a common name for an emerging family of approaches that infer the model parameters when the likelihood is intractable. Existing SBI methods either approximate the likelihood, such as Approximate Bayesian Computation (ABC), or directly model the posterior, such as Sequential Neural Posterior Estimation (SNPE). While ABC is efficient on low-dimensional problems, on higher-dimensional tasks, it is generally outperformed by SNPE which leverages function approximation. In this paper, we propose Pseudo-Likelihood Inference (PLI), a new method that brings neural approximation into ABC, making it competitive on challenging Bayesian system identification tasks. By utilizing integral probability metrics, we introduce a smooth likelihood kernel with an adaptive bandwidth that is updated based on information-theoretic trust regions. Thanks to this formulation, our method (i) allows for optimizing neural posteriors via gradient descent, (ii) does not rely on summary statistics, and (iii) enables multiple observations as input. In comparison to SNPE, it leads to improved performance when more data is available. The effectiveness of PLI is evaluated on four classical SBI benchmark tasks and on a highly dynamic physical system, showing particular advantages on stochastic simulations and multi-modal posterior landscapes.
Parametric stochastic simulators are a well-established tool for predicting the behavior of real-world phenomena. These statistical models find widespread use in various scientific fields such as physics, economics, biology, ecology, computer science, and robotics, where they help to gain knowledge about the underlying stochastic processes [27, 34] or generate additional data for subsequent downstream tasks [38]. In both cases, the practitioner seeks to explain the observations as accurately as possible while incorporating all available information. The output of such a simulator is largely determined by its parameters and their values. When estimating these parameters using Bayesian inference, given observations from a physical system, which are inevitably subject to measurement noise, we obtain a distribution over values instead of a point estimate. Additionally, there might be several parameter configurations yielding the same observation, hence rendering the resulting distribution to be multimodal. Moreover, the likelihood function might be unknown or too expensive to evaluate for many practical use cases. The combination of these difficulties makes obtaining a posterior distribution over simulator parameters challenging for state-of-the-art inference methods, both regarding effectiveness and efficiency.
SBI approaches address the issue of intractable likelihoods by using (stochastic) simulators as forward models to generate observations from proposal distributions over parameters. The approaches are also often called likelihood-free, which can be easily misunderstood since some of them directly approximate the likelihood [7]. ABC is a family of SBI methods that approximate the posterior with
a set of weighted particles which are obtained from Monte Carlo simulations and updated based on an empirical estimation of the intractable likelihood [48]. For an ABC approach to work well, three criteria have to be fulfilled: (i) the likelihood kernel is capable of measuring the similarity of observations meaningfully, (ii) the proposal distribution samples close to the posterior, (iii) the decision-making rule balances between accepting a sufficient amount of samples from the proposal and steering inference towards the posterior distribution. Constructing a suitable likelihood kernel often means tailoring summary statistics to the problem at hand. However, recent advances promise to replace this heuristic-based process by employing Integral Probability Metrics (IPMs) to measure statistical distances in observation space [3, 16, 17]. While these methods significantly increase the required number of simulations, approximations of the statistical distances [25, 8] can be computed in parallel, hence facilitating the parallelization of the whole inference pipeline. Following up on the shortcomings of ABC, the family of SNPE approaches provide Bayesian inference methods that leverage conditional neural density estimators to approximate the posterior [40, 24, 41, 19]. The benchmarking study of Lueckmann et al. [32] concludes that, generally, SNPE approaches are to be preferred over ABC as they are superior in terms of expressibility and accuracy across a wide range of benchmarking tasks. However, it is important to point out that the analysis of the posterior inference has solely been reported for single observations. These single-sample scenarios favor SNPE in high dimensions since ABC relies on summary statistics to evaluate the likelihood. Therefore, it remains an open question whether SNPE methods can transfer their benefits to settings where the (approximated) posterior is conditioned on multiple observations at once.
Contributions. By deriving the ABC posterior from a constrained variational inference objective, we introduce a novel SBI method called Pseudo-Likelihood Inference (PLI). Our formulation allows for approximating the posterior with neural density estimators. PLI updates this posterior from pseudo-likelihoods which are exponentially transformed statistical distances computed using IPMs. To further remove heuristics from the inference process, we derive an adaptive bandwidth update of PLI's likelihood kernel that bounds the loss of information based on information-geometric trust-region principles. This way, PLI can update its neural posterior solely given observations from a (stochastic) black-box simulator. Moreover, the usage of IPMs enables PLI to simultaneously condition on a variable number of observations, while SNPE methods need to concatenate them and, therefore, degrade when the number of observations increases. We compare PLI against ABC and one SNPE method on two SBI benchmarking tasks as well as a highly dynamical double pendulum task. For both, ABC and PLI, we investigate two IPMs: the Maximum Mean Discrepancy (MMD) and the Wasserstein distance. Motivated by recent benchmarking results, we chose Automatic Posterior Transformation (APT) [24] to represent SNPE approaches. Our experiments investigate the dependency of the trained density estimator's performance on the number of observations, where performance is measured in observation as well as in parameter space. We show the merits and disadvantages of all methods and conclude with concrete recommendations.
2 Bayesian inference with intractable likelihoods
The objective of Bayesian inference is to find the posterior parameter distribution  $p(\pmb{\xi}|\mathbf{x}_{1:N}^{\star})$  given a set of reference data points  $\mathbf{x}_{1:N}^{\star}$  which are assumed to be drawn from the likelihood model  $p(\mathbf{x}|\pmb{\xi})$ . Given a prior belief over the parameters  $p(\pmb{\xi})$ , the posterior is expressed via Bayes' rule
$$
p (\boldsymbol {\xi} | \mathbf {x} _ {1: N} ^ {\star}) \propto p \left(\mathbf {x} _ {1: N} ^ {\star} | \boldsymbol {\xi}\right) p (\boldsymbol {\xi}). \tag {1}
$$
In the following, we describe SBI methodologies that aim to approximate the posterior (1) when the likelihood is given by a simulator model, from which only sampling is possible,  $\mathbf{x}_{1:M} \sim p(\mathbf{x}|\pmb{\xi})$  but evaluating the likelihood is infeasible.
2.1 Approximate Bayesian computation
ABC methods perform Bayesian inference without explicitly computing the likelihood function  $p(\mathbf{x}_{1:N}^{\star}|\pmb {\xi}) = \int p(\mathbf{x}_{1:N}^{\star}|\mathbf{x}_{1:M},\pmb {\xi})p(\mathbf{x}_{1:M}|\pmb {\xi})  \mathrm{d}\mathbf{x}_{1:M}$  [48]. Instead, they approximate it
$$
\tilde {p} _ {\beta} \left(\mathbf {x} _ {1: N} ^ {\star} \mid \boldsymbol {\xi}\right) \propto \int K _ {\beta} \left(D \left(\mathbf {x} _ {1: N} ^ {\star}, \mathbf {x} _ {1: M}\right)\right) p \left(\mathbf {x} _ {1: M} \mid \boldsymbol {\xi}\right) \mathrm {d} \mathbf {x} _ {1: M} \tag {2}
$$
using Monte Carlo samples  $\mathbf{x}_{1:M}$  from the simulator as the reference points and smoothening them with a kernel  $K_{\beta}(D(\mathbf{x}_{1:N}^{\star},\mathbf{x}_{1:M}))$  [28]. The kernel assesses the similarity of the reference data
$\mathbf{x}_{1:N}^{\star}$  and the simulated data  $\mathbf{x}_{1:M}$  based on a distance measure  $D$ , the kernel type  $K$ , and the kernel bandwidth  $\beta$ . The uniform kernel  $\mathbb{1}_{\{D(\mathbf{x}_{1:N}^{\star}, \mathbf{x}_{1:M}) \leq \beta\}}$  (see Table A.1) has emerged as the default kernel choice of many ABC methods [49, 15, 30, 31]. In this case, the bandwidth represents a rejection threshold that assigns zero probability to all parameters whose simulations lie outside of the  $\beta$ -ball in terms of the distance  $D$ . The uniform kernel exhibits the favorable characteristic of converging to the likelihood in the limit [28]
$$
\lim  _ {\beta \rightarrow 0} \tilde {p} _ {\beta} \left(\mathbf {x} _ {1: N} ^ {\star} \mid \boldsymbol {\xi}\right) = \int \mathbb {1} _ {\left\{\mathbf {x} _ {1: N} ^ {\star} \right\}} \left(\mathbf {x} _ {1: M}\right) p \left(\mathbf {x} _ {1: M} \mid \boldsymbol {\xi}\right) \mathrm {d} \mathbf {x} _ {1: M} = p \left(\mathbf {x} _ {1: N} ^ {\star} \mid \boldsymbol {\xi}\right). \tag {3}
$$
Once the approximate likelihood (2) is obtained, ABC draws samples from the approximate posterior
$$
\tilde {p} _ {\beta} \left(\boldsymbol {\xi} \mid \mathbf {x} _ {1: N} ^ {\star}\right) \propto \tilde {p} _ {\beta} \left(\mathbf {x} _ {1: N} ^ {\star} \mid \boldsymbol {\xi}\right) p (\boldsymbol {\xi}). \tag {4}
$$
There exist multiple ways of implementing this sampling procedure. In rejection ABC [49], the simplest form, proposal parameters are drawn from the prior distribution  $p(\pmb{\xi})$  and are accepted if the simulated data falls close to the true data, as measured by the kernel function. While rejection ABC yields a simple algorithm with desirable convergence properties, finding posterior samples for small bandwidths  $\beta$  in high dimensions often becomes computationally infeasible [33]. Therefore, the research on ABC focuses on three directions of improvement: (i) replacing the prior with a sequentially updated proposal distribution  $\pi_t(\pmb{\xi})$  to reduce the search space during sampling, (ii) adapting the bandwidth  $\beta$  to draw samples with an appropriate acceptance rate, and (iii) finding sufficient statistics to represent the simulated output in low dimensions [48]. MCMC-ABC [33] and SMC-ABC [47, 51, 15, 31] build upon sampling strategies based on Markov Chain Monte Carlo (MCMC) and Sequential Monte Carlo (SMC) to sequentially update the proposal distribution. MCMC-ABC does not allow for an adaptive bandwidth, and thus, SMC sampling strategies have evolved as the leading ABC methods for these cases [15].
2.2 Sequential Monte Carlo ABC
SMC-ABC builds on SMC samplers introduced by Del Moral et al. [14]. Fundamentally, SMC-ABC approximates the posterior distribution through a sequence of intermediate target posterior distributions  $\tilde{p}_{\beta_t}(\boldsymbol{\xi}|\mathbf{x}_{1:N}^\star)$  (4) that are characterized by an adaptable bandwidth parameter  $\beta_{t}$ , where  $t$  denotes the inference time. Furthermore, SMC-ABC uses importance sampling from a sequentially updated proposal distribution  $\pi_t(\pmb {\xi})$  to improve the sample efficiency. The proposal distribution is represented by an empirical distribution  $\pi_t(\pmb {\xi}) = 1 / M\sum_{i = 0}^{M}\delta_{\pmb{\xi}_t^{(i)}}(\pmb {\xi})$  that is defined through a set of particles  $\{\pmb{\xi}_t^{(i)}\}$ . Importance sampling then enables the approximation of the target posterior  $\tilde{p}_{\beta_t}(\pmb {\xi}|\mathbf{x}_{1:N}^\star)$  from the proposal distribution
$$
\tilde {p} _ {\beta_ {t}} (\boldsymbol {\xi} | \mathbf {x} _ {1: N} ^ {\star}) \approx q _ {t} (\boldsymbol {\xi}) = \sum_ {i = 1} ^ {M} W _ {t} ^ {(i)} \delta_ {\boldsymbol {\xi} _ {t} ^ {(i)}} (\boldsymbol {\xi}); \quad W _ {t} ^ {(i)} = \frac {\tilde {p} _ {\beta_ {t}} \left(\boldsymbol {\xi} _ {t} ^ {(i)} \mid \mathbf {x} _ {1 : N} ^ {\star}\right)}{\pi_ {t} \left(\boldsymbol {\xi} ^ {(i)}\right)}. \tag {5}
$$
Here,  $W_{t}$  are the weights between the target posterior and the proposal distribution. SMC-ABC methods follow three steps to carry out inference for the next target posterior  $\tilde{p}_{\beta_{t + 1}}(\pmb {\xi}|\mathbf{x}_{1:N}^{\star})$  : (i) A new bandwidth  $\beta_{t + 1}$  of the target posterior  $\tilde{p}_{\beta_{t + 1}}(\mathbf{x}_{1:N}^{\star}|\pmb {\xi})$  is estimated. Typically, the update is based on heuristics, such as the Effective Sample Size (ESS) [15] to ensure that the particle variance does not degrade. (ii) New proposal particles  $\pmb{\xi}_t^{(i)}$  are sampled from a forward Markov kernel  $\pmb{\xi}_{t + 1}^{(i)}\sim K_t(\pmb{\xi}_{t - 1}^{(i)},\pmb{\xi}_t^{(i)})$  to stay close to the target posterior of the next iteration  $\tilde{p}_{\beta_{t + 1}}(\pmb {\xi}|\mathbf{x}_{1:N}^{\star})$  . (iii) The weights of the particles are adjusted based on approximations of (5). As the weight update is typically numerically intractable [14], different SMC-ABC methods [15, 30, 31] have been introduced which propose approximations to the optimal weight update. We refer to Appendix B for a more detailed explanation of SMC-ABC and its different approaches.
3 Pseudo-likelihood inference
The proposed PLI methodology, summarized in Figure 1, generalizes the ABC approaches by introducing exponential likelihood kernels with adaptive bandwidth updates, which are motivated from a Variational Inference (VI) perspective.
Figure 1: Schematic overview of the introduced iterative Pseudo-Likelihood Inference (PLI) approach. (top) Based on samples drawn from the simulator, the pseudo-likelihood (8) is evaluated based on the discrepancy between the empirical data-generating and likelihood distributions. The bar chart shows how the pseudo-likelihood evaluation changes for different bandwidths. (bottom) The evaluation of the pseudo-likelihood is used to estimate a target posterior  $\pi_t$  under trust-region constraints that moves from the prior distribution to the final posterior.
3.1 Exponential likelihood kernels
PLI adopts the view of SMC-ABC on approximating a smoothed target posterior  $p_t(\pmb{\xi} | \mathbf{x}_{1:N}^{\star})$ , in the following denoted by  $\pi_t(\pmb{\xi})$ , by formulating the following constrained VI problem for each inference step  $t$
$$
\pi_ {t} (\boldsymbol {\xi}) = \underset {\pi (\boldsymbol {\xi}) \in \mathcal {P} (\boldsymbol {\xi})} {\operatorname {a r g m i n}} \mathrm {K L} (\pi (\boldsymbol {\xi}) | | p (\boldsymbol {\xi} | \mathbf {x} _ {1: N} ^ {\star})), \tag {6}
$$
$$
\begin{array}{l} \text {s . t . K L} (\pi (\boldsymbol {\xi}) | | \pi_ {t - 1} (\boldsymbol {\xi})) \leq \varepsilon . \end{array}
$$
The optimization is balanced between fitting the posterior distribution  $p(\pmb{\xi}|\mathbf{x}_{1:N}^{\star})$  and constraining the information loss between two inference steps  $\pi_{t-1}(\pmb{\xi})$  and  $\pi_t(\pmb{\xi})$ . The loss of information is incorporated as a trust-region constraint with the bound  $\varepsilon > 0$  in the space of probability distributions  $\mathcal{P}(\pmb{\xi})$  through the Kullback-Leibler (KL) divergence  $\mathrm{KL}(\pi(\pmb{\xi}) \mid \pi_{t-1}(\pmb{\xi}))$ .
Theorem 1. The optimal target distribution  $\pi_t(\pmb{\xi})$  in the optimization problem (6) is given by
$$
\pi_ {t} (\boldsymbol {\xi}) \propto \left(\frac {p (\boldsymbol {\xi})}{\pi_ {t - 1} (\boldsymbol {\xi})}\right) ^ {\frac {1}{1 + \eta_ {t}}} p \left(\mathbf {x} _ {1: N} ^ {\star} \mid \boldsymbol {\xi}\right) ^ {\frac {1}{1 + \eta_ {t}}} \pi_ {t - 1} (\boldsymbol {\xi}) \tag {7}
$$
where  $\eta_t > 0$  is a dual Lagrangian variable corresponding to the trust-region constraint.
The temperature parameter  $\eta_{t}$  plays the role of an adaptive step size that controls the update step from  $\pi_{t-1}(\pmb{\xi})$  to  $\pi_{t}(\pmb{\xi})$ . In the limit of small step sizes  $\eta_{t} \to 0$  at convergence, the target posterior (7) turns into the true posterior  $\pi_{t}(\pmb{\xi}) \to p(\pmb{\xi}) p(\mathbf{x}_{1:N}^{\star}|\pmb{\xi})$ . In the spirit of ABC-based methods, we approximate the intractable likelihood  $p(\mathbf{x}_{1:N}^{\star}|\pmb{\xi})$  with a Gibbs distribution  $\tilde{p}(\mathbf{x}_{1:N}^{\star}|\pmb{\xi})$ , which we call pseudo-likelihood, and which is based on a discrepancy measure between the empirical data distribution  $p^{\star}(\mathbf{x}) = 1/N \sum_{i} \delta_{\mathbf{x}_i^{\star}}(\mathbf{x})$  and the simulator likelihood  $p(\mathbf{x}|\pmb{\xi})$
$$
\tilde {p} \left(\mathbf {x} _ {1: N} ^ {\star} \mid \boldsymbol {\xi}\right) := \frac {1}{Z (\boldsymbol {\xi})} \exp \left(- \frac {D \left(p ^ {\star} (\mathbf {x}) , p (\mathbf {x} \mid \boldsymbol {\xi})\right)}{2 \beta}\right). \tag {8}
$$
Here,  $Z(\pmb{\xi})$  is the normalization constant, and  $\beta > 0$  is a bandwidth parameter that controls the sharpness of the approximation. When the KL divergence is used as the discrepancy measure  $D$ , we recover the true likelihood, as the following lemma states.
Lemma 1. When  $D = \mathrm{KL}$  and  $2\beta = 1 / N$  in (8), the pseudo-likelihood  $\tilde{p}(\mathbf{x}_{1:N}^{\star}|\boldsymbol{\xi})$  equals the true likelihood  $p(\mathbf{x}_{1:N}^{\star}|\boldsymbol{\xi})$ .
Proof. Since  $\mathrm{KL}(p^{*}(\mathbf{x})\parallel p(\mathbf{x}|\boldsymbol {\xi})) = -\mathrm{H}[p^{*}(x)] - N^{-1}\log p(\mathbf{x}_{1:N}^{\star}|\boldsymbol {\xi})$ , then provided  $2\beta = 1 / N$ , the exponential in (8) is given by  $\exp (-N\mathrm{KL}(p^{*}(\mathbf{x})\parallel p(\mathbf{x}|\boldsymbol {\xi})))\propto p(\mathbf{x}_{1:N}^{\star}|\boldsymbol {\xi})$  and  $Z(\pmb {\xi})$  is a constant.
However, the KL divergence is intractable in SBI because we cannot evaluate the likelihood. Therefore, we propose replacing the KL divergence with Integral Probability Metrics (IPMs), such as the MMD and the Wasserstein distance, which can be evaluated on distribution samples. Although theoretical analysis is less straightforward in these cases, some results have been obtained in prior works. Consistency and robustness of an MMD-based posterior estimator were shown by Chérief-Abdellatif and Alquier [5] and Wasserstein-based exponential kernels were studied by De Plaen et al. [12]. In this paper, we focus on a practical instantiation of pseudo-likelihood inference, which can accommodate a variety of divergence functions and obtain superior empirical results by leveraging neural posterior approximators and adaptive step-size updates. The following subsections introduce the key components that constitute our method.
3.2 Bandwidth adaptation from trust-region principles
The Lagrangian parameter  $\eta_t$  has a particularly interesting property. From (7), we see that pulling  $\eta_t$  into the pseudo-likelihood (2), yields a time-dependent tempered pseudo-likelihood
$$
\tilde {p} _ {t} \left(\mathbf {x} _ {1: N} ^ {\star} \mid \boldsymbol {\xi}\right) := Z (\boldsymbol {\xi}) ^ {1 + \eta_ {t}} \exp \left(- \left(2 \beta_ {t}\right) ^ {- 1} D \left(p ^ {\star} (\mathbf {x}), p (\mathbf {x} \mid \boldsymbol {\xi})\right)\right) = \tilde {p} \left(\mathbf {x} _ {1: N} ^ {\star} \mid \boldsymbol {\xi}\right) ^ {\frac {1}{1 + \eta_ {t}}}. \tag {9}
$$
Here, we introduce the adaptive bandwidth  $\beta_{t} = (1 + \eta_{t})\beta$  that approaches  $\beta$  in the limit  $\eta_t\to 0$ . The dual formulation of the stochastic search problem (6) leads to a tractable solution for the optimal Lagrangian parameter  $\eta_t$ , and hence an optimal bandwidth  $\beta_{t}$  (see Appendix A.1 for more details).
$$
g \left(\eta_ {t}\right) = - \eta_ {t} \epsilon - \left(1 + \eta_ {t}\right) \log \mathbb {E} _ {\pi_ {t - 1} (\boldsymbol {\xi})} \left[ \left(\frac {p (\boldsymbol {\xi})}{\pi_ {t - 1} (\boldsymbol {\xi})}\right) ^ {\frac {1}{1 + \eta_ {t}}} \tilde {p} _ {t} \left(\mathbf {x} _ {1: N} ^ {\star} | \boldsymbol {\xi}\right) \right]. \tag {10}
$$
While we obtain the primal optimal point in closed form (7) to obtain the optimal dual variable  $\eta_t$ , we need to resort to numerical optimization of the Lagrangian dual objective. The optimal bandwidth parameter  $\beta_t$ , that is obtained by maximizing (10), can be seen as an information-bounded trust region update to move the pseudo-likelihood towards the likelihood. In the early inference stages, the proposal prior  $p_{t - 1}(\pmb {\xi})$  is typically uninformative, and thus the information loss is moderate even if  $p_t(\pmb {\xi})$  moves far away from  $p_{t - 1}(\pmb {\xi})$ . In the later inference steps, the proposal distribution is typically pronounced, and small deviations may lead to significant information loss. This intuition suggests that the bandwidth  $\beta_{t}$  should decay over iterations, and indeed Figure 2 shows that  $\beta_{t}$  quickly decays towards zero over a range of values of  $\varepsilon$  on the Gaussian location task (Sec.
C.2). The exact decay schedule of  $\beta_{t}$  is problem-dependent. Therefore, it is convenient to set an information-loss bound  $\varepsilon$  and obtain an adaptive bandwidth schedule by optimizing the dual (10) rather than pre-specifying a decay schedule by hand for each problem.
Figure 2: Bandwidth  $\eta_{t}$  recorded for different  $\varepsilon$  on the Gaussian location task. The bandwidth is monotonically decreasing over iterations.
3.3 Approximate Bayesian inference with pseudo-likelihoods
Pseudo-Likelihood Inference (PLI) is a sequential SBI methodology based on approximating the target posterior  $\pi_t(\xi)$  (7). It is closely tied to SMC-ABC by (i) sequentially approximating the target posteriors, (ii) sequentially adapting the bandwidth parameter, and (iii) sequentially updating the proposal distribution for higher sample efficiency. Instead of representing the posterior through a set of weighted particles, the PLI formulation allows for various powerful neural density estimators.
A parameterized density model  $q_{\phi}(\pmb{\xi})$  is trained to approximate the PLI posterior (7) using the m-projection  $\min_{\phi \in \Phi}\mathrm{KL}(\pi_t(\pmb {\xi})\parallel q_\phi (\pmb {\xi}))$ , which results in the Weighted Maximum Likelihood (WML) objective with parameter samples drawn from the proposal prior  $\pmb{\xi}^{(k)}\sim \pi_{t - 1}(\pmb{\xi}^{(k)})$
$$
\max  _ {\boldsymbol {\phi} \in \Phi} \sum_ {k = 1} ^ {K} w ^ {(k)} \log q _ {\boldsymbol {\phi}} (\boldsymbol {\xi} ^ {(k)}); \quad w ^ {(k)} = \left(\frac {p (\boldsymbol {\xi} ^ {(k)})}{\pi_ {t - 1} (\boldsymbol {\xi} ^ {(k)})}\right) ^ {\frac {1}{1 + \eta_ {t}}} \tilde {p} _ {t} \left(\mathbf {x} _ {1: N} ^ {\star} \mid \boldsymbol {\xi} ^ {(k)}\right). \tag {11}
$$
Thus, we derive a practical PLI algorithm by leveraging this empirical WML objective. Further details on the objective derivation are described in Appendix A.2. Our proposed PLI Algorithm 1 consists of four main steps. First, in lines 4-8, training pairs from the proposal and simulator are drawn, and the discrepancy measure  $D(\mathbf{x}_{1:N}^{\star}, \mathbf{x}_{1:M})$  between the observations and the simulations is evaluated for each  $\pmb{\xi}^{(k)}$ . We follow Gretton et al. [25] to approximate the MMD between two discrete probability measures, whereas we make use of the entropy regularized optimal transport formulation to approximate the Wasserstein distance [8] (see Appendix C.1). Both versions facilitate parallelization on the GPU. Second, in line 9, the optimal bandwidth under trust region constraint is estimated, and the tempered pseudo-likelihood is evaluated. by maximizing the dual (10). Third, in line 11, the parameterized