Geometry and Dynamics for Markov Chain Monte Carlo
It addresses the problem of making advanced geometric concepts in MCMC accessible to applied scientists like statisticians and machine learners, but it is incremental as a review.
This review tackles the gap between users' practical intuitions and the deep theoretical foundations of Hamiltonian Monte Carlo by providing a comprehensive introduction to geometric tools at an accessible level, complemented by recent advances.
Markov Chain Monte Carlo methods have revolutionised 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 which can explore probability densities efficiently. The method emerges from physics and geometry and these links have been extensively studied by a series of authors through the last thirty years. However, there is currently a gap between the intuitions and knowledge of users of the methodology and our deep understanding of these theoretical foundations. 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 applied scientists.