Likelihood-Free Adaptive Bayesian Inference via Nonparametric Distribution Matching

TL;DR

This work introduces Adaptive Bayesian Inference (ABI), a likelihood-free framework that bypasses observation-space discrepancies by matching posteriors directly in posterior space using the Marginally-augmented Sliced Wasserstein (MSW) distance. By leveraging a quantile-regression approach to estimate trimmed MSW and a sequential, generative-model–driven rejection sampling scheme, ABI achieves efficient, high-dimensional posterior inference without requiring explicit prior density evaluation. Theoretical results establish that MSW is topologically equivalent to Wasserstein, with parametric convergence rates for the trimmed version and ABI posterior convergence as the tolerance vanishes. Empirically, ABI outperforms data-space ABC variants and state-of-the-art likelihood-free simulators across a range of challenging models, including multimodal and dependent-observation settings, demonstrating robust performance and scalability. The framework opens avenues for integrating posterior-space discrepancy measures into broader sequential inference algorithms and complex simulator-based applications.

Abstract

When the likelihood is analytically unavailable and computationally intractable, approximate Bayesian computation (ABC) has emerged as a widely used methodology for approximate posterior inference; however, it suffers from severe computational inefficiency in high-dimensional settings or under diffuse priors. To overcome these limitations, we propose Adaptive Bayesian Inference (ABI), a framework that bypasses traditional data-space discrepancies and instead compares distributions directly in posterior space through nonparametric distribution matching. By leveraging a novel Marginally-augmented Sliced Wasserstein (MSW) distance on posterior measures and exploiting its quantile representation, ABI transforms the challenging problem of measuring divergence between posterior distributions into a tractable sequence of one-dimensional conditional quantile regression tasks. Moreover, we introduce a new adaptive rejection sampling scheme that iteratively refines the posterior approximation by updating the proposal distribution via generative density estimation. Theoretically, we establish parametric convergence rates for the trimmed MSW distance and prove that the ABI posterior converges to the true posterior as the tolerance threshold vanishes. Through extensive empirical evaluation, we demonstrate that ABI significantly outperforms data-based Wasserstein ABC, summary-based ABC, and state-of-the-art likelihood-free simulators, especially in high-dimensional or dependent observation regimes.

Figures (11)

• Figure 1: Comparison of approximate posterior densities obtained from ABI and alternative benchmark methods under the Multimodal Gaussian model. The true posterior is shown in the black dashed line. ABI produced posteriors that accurately align with the true posterior distribution.
• Figure 2: Comparison of marginal posteriors generated by ABI and WGAN-GP. The true posterior is shown in the black dashed line.
• Figure 3: Evolution of the sample path over successive iterations of ABI.
• Figure 4: Comparison of approximate posterior densities under the M/G/1 queuing example. The dashed black line indicates the true posterior mean at $(3.96, \,2.99, \,0.177)$. ABI outperforms alternative approaches and exhibits superior alignment with the true posterior mean.
• Figure 5: 30 trajectories simulated from the cosine model with parameter values $\omega^* = 1/80$, $\phi^* = \pi/4$, $\log(\sigma^*) = 0$, and $\log(A^*) = \log(2)$.

Theorems & Definitions (56)

• Definition 2.1: Bayes Sufficient
• Theorem 2.1: Minimal Bayes Sufficiency of Posterior Distribution
• Definition 2.2
• Remark 2.1
• Remark 2.2
• Definition 2.3: Quantile Representation of MSW Distance
• Remark 2.3
• Remark 2.4
• Definition 2.4: Deep Neural Networks
• Remark 2.5