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.

翻译：Markov 链条 Monte Carlo是巴伊西亚统计中一个重要的计算工具,但监测迭代随机算法的趋同可能具有挑战性。 在本文中,我们显示Gelman和Rubin(1992年)的趋同诊断值有严重缺陷。当链条的尾巴很重或链条差异不同时,传统美元将无法正确诊断趋同失败。在本文中,我们建议了一种基于等级的替代诊断方法来解决这些问题。我们还引入了基于量化的本地效率措施,以及计算蒙特卡洛对量化的误差估计数的实用方法。我们建议用多个链条的等级图取代共同的追踪图。 最后,我们建议如何在实践中使用这些方法。