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

Recent developments in Markov chain Monte Carlo (MCMC) algorithms now allow us to run thousands of chains in parallel almost as quickly as a single chain, using hardware accelerators such as GPUs. We explore the benefits of running many chains, with an emphasis on achieving a target precision in as short a time as possible. One expected advantage is that, while each chain still needs to forget its initial point during a warmup phase, the subsequent sampling phase can be almost arbitrarily short. To determine if the resulting short chains are reliable, we need to assess how close the Markov chains are to convergence to their stationary distribution. The $\hat R$ statistic is a general purpose and 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 $\hat R$. This new diagnostic works under conditions similar to $\hat R$ and completes the MCMC workflow for many GPU-friendly samplers. In addition, the proposed nesting provides theoretical insights into the utility of $\hat R$, in both classical and short-chains regimes.