Batch means and spectral variance estimators in Markov chain Monte Carlo

TL;DR

This paper compares spectral and batch means methods for estimating Monte Carlo standard errors in MCMC, establishing their consistency and providing practical recommendations based on finite-sample performance.

Contribution

It introduces conditions for strong and mean-square consistency of these estimators and offers guidance on choosing optimal batch sizes.

Findings

Spectral and batch means estimators are strongly consistent under certain conditions.

Conditions for mean-square consistency enable optimal batch size determination.

Empirical analysis provides practical recommendations for estimator use.

Abstract

Calculating a Monte Carlo standard error (MCSE) is an important step in the statistical analysis of the simulation output obtained from a Markov chain Monte Carlo experiment. An MCSE is usually based on an estimate of the variance of the asymptotic normal distribution. We consider spectral and batch means methods for estimating this variance. In particular, we establish conditions which guarantee that these estimators are strongly consistent as the simulation effort increases. In addition, for the batch means and overlapping batch means methods we establish conditions ensuring consistency in the mean-square sense which in turn allows us to calculate the optimal batch size up to a constant of proportionality. Finally, we examine the empirical finite-sample properties of spectral variance and batch means estimators and provide recommendations for practitioners.