### 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) |
---|---|

Article number | asz002 |

Pages (from-to) | 321-337 |

Number of pages | 17 |

Journal | Biometrika |

Volume | 106 |

Issue number | 2 |

DOIs | |

State | Published - Jan 1 2019 |

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### Keywords

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

### Cite this

*Biometrika*,

*106*(2), 321-337. [asz002]. https://doi.org/10.1093/biomet/asz002

**Multivariate output analysis for Markov chain Monte Carlo.** / Vats, Dootika; Flegal, James M.; Jones, Galin.

Research output: Contribution to journal › Article

*Biometrika*, vol. 106, no. 2, asz002, pp. 321-337. https://doi.org/10.1093/biomet/asz002

}

TY - JOUR

T1 - Multivariate output analysis for Markov chain Monte Carlo

AU - Vats, Dootika

AU - Flegal, James M.

AU - Jones, Galin

PY - 2019/1/1

Y1 - 2019/1/1

N2 - 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.

AB - 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.

KW - Covariance matrix estimation

KW - Effective sample size

KW - Markov chain Monte Carlo

KW - Multivariate analysis

UR - http://www.scopus.com/inward/record.url?scp=85068002370&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85068002370&partnerID=8YFLogxK

U2 - 10.1093/biomet/asz002

DO - 10.1093/biomet/asz002

M3 - Article

AN - SCOPUS:85068002370

VL - 106

SP - 321

EP - 337

JO - Biometrika

JF - Biometrika

SN - 0006-3444

IS - 2

M1 - asz002

ER -