Convergence diagnostics for Markov chain Monte Carlo

TL;DR

This paper reviews key convergence diagnostics and stopping rules for Markov chain Monte Carlo, highlighting practical tools and recent theoretically grounded methods with illustrative examples.

Contribution

It provides a comprehensive overview of empirical and theoretical convergence diagnostics and stopping rules for MCMC, including recent advances with strong theoretical support.

Findings

Widely used MCMC convergence diagnostics discussed

Recent stopping rules with theoretical guarantees presented

Illustrative examples demonstrate practical application

Abstract

Markov chain Monte Carlo (MCMC) is one of the most useful approaches to scientific computing because of its flexible construction, ease of use and generality. Indeed, MCMC is indispensable for performing Bayesian analysis. Two critical questions that MCMC practitioners need to address are where to start and when to stop the simulation. Although a great amount of research has gone into establishing convergence criteria and stopping rules with sound theoretical foundation, in practice, MCMC users often decide convergence by applying empirical diagnostic tools. This review article discusses the most widely used MCMC convergence diagnostic tools. Some recently proposed stopping rules with firm theoretical footing are also presented. The convergence diagnostics and stopping rules are illustrated using three detailed examples.