November 2021 Revisiting the Gelman–Rubin Diagnostic
Dootika Vats, Christina Knudson
Author Affiliations +
Statist. Sci. 36(4): 518-529 (November 2021). DOI: 10.1214/20-STS812


Gelman and Rubin’s (Statist. Sci. 7 (1992) 457–472) convergence diagnostic is one of the most popular methods for terminating a Markov chain Monte Carlo (MCMC) sampler. Since the seminal paper, researchers have developed sophisticated methods for estimating variance of Monte Carlo averages. We show that these estimators find immediate use in the Gelman–Rubin statistic, a connection not previously established in the literature. We incorporate these estimators to upgrade both the univariate and multivariate Gelman–Rubin statistics, leading to improved stability in MCMC termination time. An immediate advantage is that our new Gelman–Rubin statistic can be calculated for a single chain. In addition, we establish a one-to-one relationship between the Gelman–Rubin statistic and effective sample size. Leveraging this relationship, we develop a principled termination criterion for the Gelman–Rubin statistic. Finally, we demonstrate the utility of our improved diagnostic via examples.


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Dootika Vats. Christina Knudson. "Revisiting the Gelman–Rubin Diagnostic." Statist. Sci. 36 (4) 518 - 529, November 2021.


Published: November 2021
First available in Project Euclid: 11 October 2021

MathSciNet: MR4323050
zbMATH: 07473933
Digital Object Identifier: 10.1214/20-STS812

Keywords: batch means , Convergence diagnostic , effective sample size , Gelman–Rubin , Markov chain Monte Carlo

Rights: Copyright © 2021 Institute of Mathematical Statistics


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Vol.36 • No. 4 • November 2021
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