Geometry in sampling methods: A review on manifold MCMC and particle-based variational inference methods

Abstract

Sampling methods are an indispensable tool for Bayesian inference, as they provide a flexible and asymptotically exact approximation to the intractable posterior in an out-of-the-box way. These methods generate or update the samples by simulating a dynamical process, which is a construct on a space with certain geometry. Non-Euclidean geometry has long been incorporated in Bayesian inference and continues to generate impact. It is considered either (1) directly due to that the target distribution is defined on a non-Euclidean manifold, or (2) for a proper dynamics that respects the geometry of a distribution space. In this chapter, we review the background and some recent progress on the interplay between geometry and sampling methods. We consider two major classes of sampling methods: Markov chain Monte Carlo (MCMC) and particle-based variational inference (ParVI). For MCMC, we cover some dynamics on manifolds and their simulation for both cases (1) and (2). For ParVI, we describe its geometric interpretation under the view of case (2), and introduce the variants that the interpretation inspires, including those for case (1).