## Abstract

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 language | English (US) |
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Article number | asz002 |

Pages (from-to) | 321-337 |

Number of pages | 17 |

Journal | Biometrika |

Volume | 106 |

Issue number | 2 |

DOIs | |

State | Published - 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.

## Keywords

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