Metropolis-Hastings via Classification

TL;DR

This paper introduces a novel Bayesian sampling method that uses classification techniques inspired by GANs to estimate likelihood ratios, enabling efficient posterior sampling when likelihoods are hard to compute.

Contribution

It proposes a new likelihood-free Bayesian inference approach combining classification with Metropolis-Hastings, with theoretical analysis and practical demonstrations.

Findings

Effective in challenging likelihood-free scenarios

Provides asymptotic normality and convergence rate analysis

Generates approximate posterior samples with theoretical guarantees

Abstract

This paper develops a Bayesian computational platform at the interface between posterior sampling and optimization in models whose marginal likelihoods are difficult to evaluate. Inspired by adversarial optimization, namely Generative Adversarial Networks (GAN), we reframe the likelihood function estimation problem as a classification problem. Pitting a Generator, who simulates fake data, against a Classifier, who tries to distinguish them from the real data, one obtains likelihood (ratio) estimators which can be plugged into the Metropolis-Hastings algorithm. The resulting Markov chains generate, at a steady state, samples from an approximate posterior whose asymptotic properties we characterize. Drawing upon connections with empirical Bayes and Bayesian mis-specification, we quantify the convergence rate in terms of the contraction speed of the actual posterior and the convergence rate of the Classifier. Asymptotic normality results are also provided which justify inferential potential of our approach. We illustrate the usefulness of our approach on examples which have posed a challenge for existing Bayesian likelihood-free approaches.