Mixing of Metropolis-adjusted Langevin and Hamiltonian Monte Carlo Methods via Coupling

Markov chain Monte Carlo methods are algorithms designed to extract information encoded in probability distributions. By generating approximate samples along a Markov chain, they make it possible to examine such target distributions, for instance through the approximation of expectations by empirical averages. From their emergence in physics and subsequent rise to general purpose methodology in statistics, they are now widely used across countless fields of study. The Metropolis algorithm, introduced in 1953 as the first Markov chain Monte Carlo method, proposed a simple yet powerful accept–reject mechanism to construct suitable Markov chain transitions. Owing to its flexibility, it has remained a constant throughout the development of Markov chain Monte Carlo. In the context of Langevin and Hamiltonian Monte Carlo, that is, methods employing Langevin dynamics or Hamiltonian flow to efficiently explore the distribution under consideration, first introduced in the late 1980s and actively developed since, the Metropolis mechanism takes the crucial role of correcting discretization bias originating in the numerical approximation of the underlying continuous dynamics. This dissertation concerns itself with the mixing, meaning convergence to the target distribution, of Metropolis-adjusted Langevin and Hamiltonian Monte Carlo methods. Identifying and quantifying the mechanisms sustaining their convergence is crucial for understanding algorithmic performance and gives rise to a rich area of mathematical research. After reviewing their historical development in Chapter 1, we systematically study the coupling-based analysis of their convergence to equilibrium in Chapters 2 and 3, which leads to a natural approach to quantify their mixing. Based on this perspective, in the subsequent Chapter 4, we develop frameworks that reduce the analysis to fundamental sets of conditions, and that can be flexibly applied to various methods of interest. The appendix contains four independent projects, each studying the mixing of a specific method.