Nested $\hat R$: Assessing the convergence of Markov chain Monte Carlo when running many short chains

TL;DR

This paper introduces a nested convergence diagnostic called nested R for assessing the reliability of many short Markov chains run in parallel, especially suited for GPU-accelerated MCMC, improving convergence assessment efficiency.

Contribution

The paper proposes a novel nested R diagnostic that enhances convergence assessment for many short chains, addressing limitations of traditional R diagnostics in GPU-accelerated MCMC.

Findings

Nested R provides reliable convergence diagnostics for short chains.

The method offers theoretical insights into R's utility in different regimes.

It enables efficient convergence checking in parallel MCMC settings.

Abstract

Recent developments in parallel Markov chain Monte Carlo (MCMC) algorithms allow us to run thousands of chains almost as quickly as a single chain, using hardware accelerators such as GPUs. While each chain still needs to forget its initial point during a warmup phase, the subsequent sampling phase can be shorter than in classical settings, where we run only a few chains. To determine if the resulting short chains are reliable, we need to assess how close the Markov chains are to their stationary distribution after warmup. The potential scale reduction factor $\widehat R$ is a popular convergence diagnostic but unfortunately can require a long sampling phase to work well. We present a nested design to overcome this challenge and a generalization called nested $\widehat R$. This new diagnostic works under conditions similar to $\widehat R$ and completes the workflow for GPU-friendly samplers. In addition, the proposed nesting provides theoretical insights into the utility of $\widehat R$, in both classical and short-chains regimes.