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 $\widehat{R}$
of Gelman and Rubin (1992) has serious flaws. Traditional $\widehat{R}$ 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.

本文提出了一种修正了 Gelman 和 Rubin 的收敛诊断 R^ 的新方法，以解决其在链条存在重尾或方差变化等问题时不能准确诊断收敛失败的问题，并依此提出了一些基于分位数的局部效率量及其误差估计方法，并建议使用多个链条的排名图，取代传统的追踪图，并对其实际应用进行了一系列建议。