  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.
