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

The growing availability of hardware accelerators such as GPUs has generated interest in Markov chains Monte Carlo (MCMC) workflows which run a large number of chains in parallel. Each chain still needs to forget its initial state but 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 $\widehat R$ statistic is a battle-tested convergence diagnostic but unfortunately can require long chains to work well. We present a nested design to overcome this challenge, and introduce tuning parameters to control the reliability, bias, and variance of convergence diagnostics.