Multivariate output analysis for Markov chain Monte Carlo

Dootika Vats, James M. Flegal, Galin L. Jones

Research output: Contribution to journalArticlepeer-review

101 Scopus citations


Markov chain Monte Carlo produces a correlated sample which may be used 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. The multivariate nature of this Monte Carlo error has been largely ignored in the literature. We present a multivariate framework for terminating a simulation in Markov chain Monte Carlo. We define a multivariate effective sample size, the estimation of which requires strongly consistent estimators of the covariance matrix in the Markov chain central limit theorem, 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 does not depend on the underlying stochastic process and can be calculated a priori. 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.

Original languageEnglish (US)
Article numberasz002
Pages (from-to)321-337
Number of pages17
Issue number2
StatePublished - Jun 1 2019

Bibliographical note

Funding Information:
Flegal’s work was partially supported by the National Science Foundation. Jones was partially supported by the National Science Foundation and the National Institutes for Health.

Publisher Copyright:
© 2019 Biometrika Trust.


  • Covariance matrix estimation
  • Effective sample size
  • Markov chain Monte Carlo
  • Multivariate analysis


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