# Multivariate output analysis for Markov chain Monte Carlo

Dootika Vats, James M. Flegal, Galin Jones

Research output: Contribution to journalArticle

4 Citations (Scopus)

### 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) asz002 321-337 17 Biometrika 106 2 https://doi.org/10.1093/biomet/asz002 Published - Jan 1 2019

### Fingerprint

Markov Chains
Markov Chain Monte Carlo
Markov processes
Multivariate Analysis
Effective Sample Size
Output
Sample Size
Central limit theorem
Stochastic Processes
Markov chain
sampling
Batch Means
Lower bound
Covariance matrix
Random processes
Simulation
Stopping Rule
Consistent Estimator
stochastic processes
Standard deviation

### Keywords

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

### Cite this

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

In: Biometrika, Vol. 106, No. 2, asz002, 01.01.2019, p. 321-337.

Research output: Contribution to journalArticle

Vats, Dootika ; Flegal, James M. ; Jones, Galin. / Multivariate output analysis for Markov chain Monte Carlo. In: Biometrika. 2019 ; Vol. 106, No. 2. pp. 321-337.
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