Open Access
2023 On the Use of a Local Rˆ to Improve MCMC Convergence Diagnostic
Théo Moins, Julyan Arbel, Anne Dutfoy, Stéphane Girard
Author Affiliations +
Bayesian Anal. Advance Publication 1-26 (2023). DOI: 10.1214/23-BA1399


Diagnosing convergence of Markov chain Monte Carlo is crucial and remains an essentially unsolved problem. Among the most popular methods, the potential scale reduction factor, commonly named Rˆ, is an indicator that monitors the convergence of output chains to a target distribution, based on a comparison of the between- and within-variances. Several improvements have been suggested since its introduction in the 90s. Here, we aim at better understanding the Rˆ behavior by proposing a localized version that focuses on quantiles of the target distribution. This new version relies on key theoretical properties of the associated population value. It naturally leads to proposing a new indicator Rˆ, which is shown to allow both for localizing the Markov chain Monte Carlo convergence in different quantiles of the target distribution, and at the same time for handling some convergence issues not detected by other Rˆ versions.

Funding Statement

SG acknowledges the support of the Chair Stress Test, Risk Management and Financial Steering, led by the École polytechnique and its Foundation and sponsored by BNP Paribas. JA acknowledges the support of the French National Research Agency (ANR-21-JSTM-0001).


We would like to thank a Reviewer and an Editor for providing us with valuable comments that helped us improving the manuscript. Specifically, comments from an Editor allowed us to deal with multi-stage testing in an appealing and satisfactory way.


Download Citation

Théo Moins. Julyan Arbel. Anne Dutfoy. Stéphane Girard. "On the Use of a Local Rˆ to Improve MCMC Convergence Diagnostic." Bayesian Anal. Advance Publication 1 - 26, 2023.


Published: 2023
First available in Project Euclid: 18 July 2023

Digital Object Identifier: 10.1214/23-BA1399

Keywords: computational statistics , convergence diagnostics , Markov chain Monte Carlo , potential scale reduction factor

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