Metropolis–Hastings via Classification

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 likelihood function estimation problem as classification problem. Pitting Generator, who simulates fake data, against Classifier, tries distinguish them from real one obtains (ratio) estimators which can be plugged into Metropolis-Hastings algorithm. The resulting Markov chains generate, steady state, samples an approximate asymptotic properties characterize. Drawing upon connections with empirical Bayes mis-specification, quantify convergence rate terms of contraction speed actual Classifier. Asymptotic normality results also provided justify inferential potential our approach. We illustrate usefulness approach on examples have posed challenge for existing likelihood-free approaches.