Markov chain Monte Carlo in statistical mechanics: The problem of accuracy

The appearance of the article by N. Metropolis. A.W. Rosenbluth, M.N. Rosen bluth, A. H. Teller, and E. Teller marked the birth of the Monte Carlo meth od for the study of statistical-mechanical systems and of a specific form o f "importance sampling"-namely, Markov chain Monte Carlo. After nearly 40 y ears of statistical usage, this technique has had a profound impact on stat istical theory, on both Bayesian and classical statistics. Markov chain Mon te Carlo is used essentially to estimate integrals in high dimensions. This article addresses the accuracy of such estimation. Through computer experi ments performed on the two-dimensional Ising model, we compare the most com mon method for error estimates in statistical mechanics. It appears that th e moving-block bootstrap outperforms other methods based on subseries value s when the number of observations is relatively small and the time correlat ion between successive configurations decays slowly. Moreover, the moving-b lock bootstrap enables estimates of the standard error to be made not only for the averages of directly obtained data but also for estimates derived f rom sophisticated numerical procedures.