Nonasymptotic bounds on the estimation error for regenerative MCMC algorithms

TL;DR

This paper derives nonasymptotic bounds on the estimation error of regenerative MCMC algorithms, providing practical error and confidence bounds for expectations estimated from Markov chain samples.

Contribution

It introduces explicit nonasymptotic mean square error bounds for regenerative MCMC estimators, including bounds on asymptotic variance for different ergodicity conditions.

Findings

Provides inequalities for mean square error of estimators

Derives bounds on asymptotic variance for uniformly ergodic chains

Extends results to geometrically ergodic chains

Abstract

MCMC methods are used in Bayesian statistics not only to sample from posterior distributions but also to estimate expectations. Underlying functions are most often defined on a continuous state space and can be unbounded. We consider a regenerative setting and Monte Carlo estimators based on i.i.d. blocks of a Markov chain trajectory. The main result is an inequality for the mean square error. We also consider confidence bounds. We first derive the results in terms of the asymptotic variance and then bound the asymptotic variance for both uniformly ergodic and geometrically ergodic Markov chains.