Optimal batch sizes for variance estimators in MCMC

This paper proposes optimal mean squared error batch sizes for multivariate batch means and spectral variance estimators. We propose a novel estimation technique for the optimal batch sizes, which is computationally inexpensive and has low variability. Further, the asymptotic mean squared error for a family of spectral variance estimators is derived under conditions convenient to verify for Markov chain Monte Carlo simulations. Vector auto-regressive, Bayesian logistic regression, and Bayesian dynamic space-time examples illustrate the quality of the estimation procedure where optimal batch sizes proposed here outperform current batch size selection methods.