OPTIMAL VARIANCE REDUCTION FOR MARKOV CHAIN MONTE CARLO

Markov chain Monte Carlo (MCMC) has been widely used to approximate the expectation of the statistic of a given probability measure pi on a finite set, and the asymptotic variance is a typical approach to evaluating the performance of MCMC methods. In this paper, we provide a lower bound of the worst-case analysis of the asymptotic variance over general Markov chains with invariant probability pi, reversible as well as nonreversible ones, and construct an optimal transition matrix that achieves this lower bound. In fact, for any statistic f to be evaluated, an MCMC with the constructed optimal transition matrix produces a smaller asymptotic variance than independent sampling.