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Estimating accuracy of the MCMC variance estimator: a central limit theorem for batch means estimators

The batch means estimator of the MCMC variance is a simple and effective measure of accuracy for MCMC based ergodic averages. Under various regularity conditions, the estimator has been shown to be consistent for the true variance. However, the estimator can be unstable in practice as it depends directly on the raw MCMC output. A measure of accuracy of the batch means estimator itself, ideally in the form of a confidence interval, is therefore desirable. The asymptotic variance of the batch means estimator is known; however, without any knowledge of asymptotic distribution, asymptotic variances are in general insufficient to describe variability. In this article we prove a central limit theorem for the batch means estimator that allows for the construction of asymptotically accurate confidence intervals for the batch means estimator. Additionally, our results provide a Markov chain analogue of the classical CLT for the sample variance parameter for i.i.d. observations. Our result assumes standard regularity conditions similar to the ones assumed in the literature for proving consistency. Simulated and real data examples are included as illustrations and applications of the CLT.