Diagnosing convergence of Markov chain Monte Carlo is crucial and remains an essentially unsolved problem. Among the most popular methods, the potential scale reduction factor, commonly named $\hat{R}$, is an indicator that monitors the convergence of output chains to a target distribution, based on a comparison of the between- and within-variances. Several improvements have been suggested since its introduction in the 90s. Here, we aim at better understanding the $\hat{R}$ behavior by proposing a localized version that focuses on quantiles of the target distribution. This new version relies on key theoretical properties of the associated population value. It naturally leads to proposing a new indicator $\hat{R}_\infty$, which is shown to allow both for localizing the Markov chain Monte Carlo convergence in different quantiles of the target distribution, and at the same time for handling some convergence issues not detected by other $\hat{R}$ versions.

翻译：马尔可夫链蒙特卡洛的收敛诊断至关重要，但至今仍是一个尚未根本解决的问题。在最常用的方法中，潜在尺度缩减因子（通常称为$\hat{R}$）是一种通过比较链间方差与链内方差来监测输出链是否收敛于目标分布的指标。自20世纪90年代提出以来，已有多种改进方案。本文旨在通过提出一种关注目标分布分位数的局部化版本，更深入地理解$\hat{R}$的行为。该新版本基于关联总体值的关键理论性质，自然引出了新指标$\hat{R}_\infty$。结果表明，$\hat{R}_\infty$既能定位马尔可夫链蒙特卡洛在目标分布不同分位数处的收敛情况，同时又能处理其他$\hat{R}$版本未检测到的某些收敛问题。