Multivariate Output Analysis for Markov chain Monte Carlo

Markov chain Monte Carlo (MCMC) produces a correlated sample in order to estimate expectations with respect to a target distribution. A fundamental question is when should sampling stop so that we have good estimates of the desired quantities? The key to answering this question lies in assessing the Monte Carlo error through a multivariate Markov chain central limit theorem. However, the multivariate nature of this Monte Carlo error has largely been ignored in the MCMC literature. We provide conditions for consistently estimating the asymptotic covariance matrix via the multivariate batch means estimator. Based on this result, we present a relative standard deviation fixed-volume sequential stopping rule for terminating simulation. We show that this stopping rule is asymptotically equivalent to terminating when the effective sample size for the MCMC sample is above a desired threshold, giving an intuitive and theoretically justified approach to implementing the proposed method. The finite sample properties of the proposed method are demonstrated in examples.