Geometry and Dynamics for Markov Chain Monte Carlo

Abstract

Markov chain Monte Carlo methods have revolutionized mathematical computation and enabled statistical inference within many previously intractable models. In this context, Hamiltonian dynamics have been proposed as an efficient way of building chains that can explore probability densities efficiently. The method emerges from physics and geometry, and these links have been extensively studied over the past thirty years. The aim of this review is to provide a comprehensive introduction to the geometric tools used in Hamiltonian Monte Carlo at a level accessible to statisticians, machine learners, and other users of the methodology with only a basic understanding of Monte Carlo methods. This will be complemented with some discussion of the most recent advances in the field, which we believe will become increasingly relevant to scientists.