Error bounds for computing the expectation by Markov chain Monte Carlo

TL;DR

This paper derives explicit error bounds for reversible Markov chain Monte Carlo methods in approximating expectations, analyzing the impact of burn-in and providing optimal strategies to improve accuracy.

Contribution

It provides the first explicit error bounds for different norms and discusses optimal burn-in strategies for MCMC algorithms.

Findings

Error bounds match asymptotic limits

Burn-in significantly affects error reduction

Optimal burn-in strategies improve estimation accuracy

Abstract

We study the error of reversible Markov chain Monte Carlo methods for approximating the expectation of a function. Explicit error bounds with respect to different norms of the function are proven. By the estimation the well known asymptotical limit of the error is attained, i.e. there is no gap between the estimate and the asymptotical behavior. We discuss the dependence of the error on a burn-in of the Markov chain. Furthermore we suggest and justify a specific burn-in for optimizing the algorithm.