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Rank-Normalization, Folding, and Localization: An Improved Rˆ for Assessing Convergence of MCMC (with Discussion)
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Markov chain Monte Carlo is a key computational tool in Bayesian statistics, but it can be challenging to monitor the convergence of an iterative stochastic algorithm. In this paper we show that the convergence diagnostic Rˆ of Gelman and Rubin (1992) has serious flaws. Traditional Rˆ will fail to correctly diagnose convergence failures when the chain has a heavy tail or when the variance varies across the chains. In this paper we propose an alternative rank-based diagnostic that fixes these problems. We also introduce a collection of quantile-based local efficiency measures, along with a practical approach for computing Monte Carlo error estimates for quantiles. We suggest that common trace plots should be replaced with rank plots from multiple chains. Finally, we give recommendations for how these methods should be used in practice.
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Metadata
- Authors
- Vehtari, A., Gelman, A., Simpson, D., Carpenter, B., & Bürkner, P.-C.
- Affiliations
- Aalto University (FI)
- Columbia University (US)
- University of Toronto (CA)
- Flatiron Institute (US)
- Venue
- Bayesian Analysis
- Publisher
- Institute of Mathematical Statistics
- Published
- 2021-06-01
- Volume
- 16
- Issue
- 2
- Pages
- 667-718
- ISSN
- 1931-6690 (ISSN-L)
- 1936-0975
- OpenAlex
- W2920804790
- Cited by
- 1,810
- References
- 32
- Citations by year
- 2026: 499 · 2025: 424 · 2024: 328 · 2023: 263 · 2022: 153 · 2021: 93
- FWCI
- 96.56
- Type
- journal article
- Language
- en
- Open access
- diamond
- doi.org
- Licence
- cc-by
- Locations
- Bayesian Analysis (publishedVersion, cc-by)
- arXiv (Cornell University) (submittedVersion)
- HAL (submittedVersion, other-oa)
- Aaltodoc (Aalto University) (submittedVersion, cc-by)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Mathematics › Statistics and Probability › Markov Chains and Monte Carlo Methods
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Bayesian Inference
- Physical Sciences › Mathematics › Statistics and Probability › Data Analysis with R
- Keywords
- convergence diagnostics, Markov chain Monte Carlo, heavy-tailed distributions, Monte Carlo error estimation, trace plots
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
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