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General Methods for Monitoring Convergence of Iterative Simulations
Abstract
We generalize the method proposed by Gelman and Rubin (1992a) for monitoring the convergence of iterative simulations by comparing between and within variances of multiple chains, in order to obtain a family of tests for convergence. We review methods of inference from simulations in order to develop convergence-monitoring summaries that are relevant for the purposes for which the simulations are used. We recommend applying a battery of tests for mixing based on the comparison of inferences from individual sequences and from the mixture of sequences. Finally, we discuss multivariate analogues, for assessing convergence of several parameters simultaneously.
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“We ran three parallel MCMC chains of 1,000 each after burn-in time of 1,000 iterations, which was sufficient to ensure convergence, as judged by visual inspection of the chain histories and the Gelman–Rubin statistic (Rhat = 1.0 for all parameters in all models; ref. 81).”
From Active learning narrows achievement gaps for underrepresented students in undergraduate science, technology, engineering, and math (2020) · cites this paper as [81]
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“To ensure convergence, trace plots and the BrooksGelmanRubin statistic were assessed.24”
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Metadata
- Authors
- Brooks, S. P., & Gelman, A.
- Affiliations
- University of Bristol (GB)
- Columbia University (US)
- Venue
- Journal of Computational and Graphical Statistics
- Publisher
- Informa UK Limited
- Published
- 1998-12
- Volume
- 7
- Issue
- 4
- Pages
- 434-455
- ISSN
- 1061-8600 (ISSN-L)
- 1537-2715
- OpenAlex
- W2093223772
- Cited by
- 6,045
- References
- 28
- Citations by year
- 2026: 321 · 2025: 371 · 2024: 394 · 2023: 381 · 2022: 341 · 2021: 445 · 2020: 435 · 2019: 370 · 2018: 329 · 2017: 276 · 2016: 305 · 2015: 300 · 2014: 250 · 2013: 244 · 2012: 249
- FWCI
- 15.55
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- Journal of Computational and Graphical Statistics (publishedVersion)
- Bristol Research (University of Bristol) (submittedVersion)
- Funders
- National Science Foundation
- KU Leuven
- Engineering and Physical Sciences Research Council
- Topics
- Physical Sciences › Mathematics › Statistics and Probability › Markov Chains and Monte Carlo Methods
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Bayesian Inference
- Social Sciences › Decision Sciences › Management Science and Operations Research › Simulation Techniques and Applications
- Keywords
- convergence monitoring, Markov chain Monte Carlo, mixing test, simulation-based inference
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.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
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