Convergence Bounds for Monte Carlo Markov Chains

TL;DR

This paper reviews methods for analyzing the convergence of Markov Chain Monte Carlo algorithms, focusing on coupling and L2 techniques, and discusses strategies for detecting slow convergence.

Contribution

It provides a comprehensive overview of convergence bounds in MCMC, emphasizing coupling and conductance-based methods, for researchers new to the field.

Findings

Explains coupling and L2 convergence methods.

Highlights conductance and isoperimetric inequalities.

Discusses strategies for identifying slow convergence.

Abstract

This review paper, written for the second edition of the Handbook of Markov Chain Monte Carlo, provides an introduction to the study of convergence analysis for Markov chain Monte Carlo (MCMC), aimed at researchers new to the field. We focus on methods for constructing bounds on the distance between the distribution of a Markov chain at a given time and its stationary distribution. Two widely-used approaches are explored: the coupling method and the L2 theory of Markov chains. For the latter, we emphasize techniques based on conductance and isoperimetric inequalities. Additionally, we briefly discuss strategies for identifying slow convergence in Markov chains.