Convergence Speed and Approximation Accuracy of Numerical MCMC

TL;DR

This paper analyzes how numerical errors in MCMC algorithms influence their convergence speed and estimation accuracy, providing theoretical bounds and practical insights for common sampling methods.

Contribution

It offers a comprehensive analysis of the effects of perturbations on MCMC convergence and accuracy, extending existing frameworks and applying results to various algorithms.

Findings

Perturbed MCMC maintains similar convergence speed when approximation is good.

High approximation accuracy is achievable under certain conditions.

Numerical examples confirm theoretical predictions.

Abstract

When implementing Markov Chain Monte Carlo (MCMC) algorithms, perturbation caused by numerical errors is sometimes inevitable. This paper studies how perturbation of MCMC affects the convergence speed and Monte Carlo estimation accuracy. Our results show that when the original Markov chain converges to stationarity fast enough and the perturbed transition kernel is a good approximation to the original transition kernel, the corresponding perturbed sampler has similar convergence speed and high approximation accuracy as well. We discuss two different analysis frameworks: ergodicity and spectral gap, both are widely used in the literature. Our results can be easily extended to obtain non-asymptotic error bounds for MCMC estimators. We also demonstrate how to apply our convergence and approximation results to the analysis of specific sampling algorithms, including Random walk Metropolis and Metropolis adjusted Langevin algorithm with perturbed target densities, and parallel tempering Monte Carlo with perturbed densities. Finally we present some simple numerical examples to verify our theoretical claims.