Some advances in proximal MCMC

Data Science Seminar

Eric Chi
University of Minnesota School of Statistics

Abstract

Markov chain Monte Carlo (MCMC) is a workhorse computational framework for sampling from complex target distributions. When targets are sufficiently smooth, gradient-based MCMC methods can more efficiently explore the target space. In the context of Bayesian inverse problems, however, targets of interest are often non-differentiable, limiting the use of gradient-based approaches. To address this limitation, Proximal MCMC methods use the Moreau–Yosida (MY) envelope to smooth non-smooth targets, enabling gradient-based sampling while preserving the critical structure of the original non-smooth target. We discuss two recent extensions of proximal MCMC. First, MCMC Importance Sampling via MY envelopes uses the envelope as an importance distribution, yielding lower-variance estimators. Second, Proximal Hamiltonian Monte Carlo (p-HMC) applies MY envelopes in an analogous fashion to forward-backward splitting in HMC sampling, thereby improving gradient approximations for non-smooth posteriors. We illustrate how these two extensions enable proximal MCMC to sample posteriors more efficiently in Bayesian models, offering both theoretical guarantees and practical performance gains.