Estimating MCMC convergence rates using common random number simulation

TL;DR

This paper introduces a CRN-based method to evaluate MCMC convergence rates, providing tighter bounds on the Wasserstein distance and demonstrating faster convergence in practical models.

Contribution

It develops a novel CRN simulation approach to bound MCMC convergence, outperforming traditional bounds in specific models.

Findings

CRN-based bounds converge faster than traditional bounds

Effective in Gibbs samplers for complex models

Provides practical tools for assessing MCMC convergence

Abstract

This paper presents how to use common random number (CRN) simulation to evaluate Markov chain Monte Carlo (MCMC) convergence to stationarity. We provide an upper bound on the Wasserstein distance of a Markov chain to its stationary distribution after $N$ steps in terms of averages over CRN simulations. We apply our bound to Gibbs samplers on a model related to James-Stein estimators, a variance component model, and a Bayesian linear regression model. For the first two examples, we show that the CRN simulated bound converges to zero significantly more quickly compared to available drift and minorization bounds.