Nonasymptotic bounds on the mean square error for MCMC estimates via renewal techniques

TL;DR

This paper derives nonasymptotic bounds on the mean square error of regenerative MCMC estimates using renewal theory, providing practical confidence intervals and comparing different schemes under specific chain conditions.

Contribution

It introduces new nonasymptotic bounds for MCMC estimates based on renewal techniques, applicable to chains satisfying Doeblin or geometric drift conditions.

Findings

Bounds improve understanding of MCMC estimation accuracy

Explicit bounds enable better confidence interval construction

Comparison of schemes under different chain conditions

Abstract

The Nummellin's split chain construction allows to decompose a Markov chain Monte Carlo (MCMC) trajectory into i.i.d. "excursions". RegenerativeMCMC algorithms based on this technique use a random number of samples. They have been proposed as a promising alternative to usual fixed length simulation [25, 33, 14]. In this note we derive nonasymptotic bounds on the mean square error (MSE) of regenerative MCMC estimates via techniques of renewal theory and sequential statistics. These results are applied to costruct confidence intervals. We then focus on two cases of particular interest: chains satisfying the Doeblin condition and a geometric drift condition. Available explicit nonasymptotic results are compared for different schemes of MCMC simulation.