2015/12/24 by Dootika Vats, James M Flegal, James M. Flegal +5 · 47 citations
Mathematics · #Applied mathematics #Estimator #Markov Chains and Monte Carlo Methods #Markov chain #Markov chain Monte Carlo #Mathematical optimization #Mathematics #Monte Carlo method #Multivariate normal distribution #Multivariate statistics #Statistical Methods and Bayesian Inference #Statistical Methods and Inference #Statistics #math.ST #stat.CO #stat.TH
paper · pdf · doi:10.1093/biomet/asz002
published in Biometrika 106(2), 321-337 (Oxford University Press)
arxiv created 2017/09/29 · arxiv updated 2017/10/02 · openalex publication_date 2019/01/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Markov chain Monte Carlo (MCMC) produces a correlated sample for estimating 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 (CLT). The multivariate nature of this Monte Carlo error largely has been ignored in the MCMC literature. We present a multivariate framework for terminating simulation in MCMC. We define a multivariate effective sample size, estimating which requires strongly consistent estimators of the covariance matrix in the Markov chain CLT; a property we show for the multivariate batch means estimator. We then provide a lower bound on the number of minimum effective samples required for a desired level of precision. This lower bound depends on the problem only in the dimension of the expectation being estimated, and not on the underlying stochastic process. This result is obtained by drawing a connection between terminating simulation via effective sample size and terminating simulation using a relative standard deviation fixed-volume sequential stopping rule; which we demonstrate is an asymptotically valid procedure. The finite sample properties of the proposed method are demonstrated in a variety of examples.