Rank-Normalization, Folding, and Localization: An Improved Rˆ for Assessing Convergence of MCMC (with Discussion)

Abstract

Markov chain Monte Carlo is a key computational tool in Bayesian statistics,\nbut it can be challenging to monitor the convergence of an iterative stochastic\nalgorithm. In this paper we show that the convergence diagnostic $\\widehat{R}$\nof Gelman and Rubin (1992) has serious flaws. Traditional $\\widehat{R}$ will\nfail to correctly diagnose convergence failures when the chain has a heavy tail\nor when the variance varies across the chains. In this paper we propose an\nalternative rank-based diagnostic that fixes these problems. We also introduce\na collection of quantile-based local efficiency measures, along with a\npractical approach for computing Monte Carlo error estimates for quantiles. We\nsuggest that common trace plots should be replaced with rank plots from\nmultiple chains. Finally, we give recommendations for how these methods should\nbe used in practice.\n

Affiliated Institutions

• Columbia University US
• Aalto University FI
• University of Toronto CA
• Flatiron Institute
• Flatiron Health (United States) US

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