On the use of a local hatR to improve MCMC convergence diagnostic

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 hatR, 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 hatR 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 hatR_∞, 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 hatR versions.