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.

翻译：并行马尔可夫链蒙特卡洛（MCMC）算法的最新进展使我们能够利用GPU等硬件加速器，以几乎与单条链相当的速度运行数千条链。尽管每条链在预热阶段仍需遗忘其初始点，但随后的采样阶段可比经典设置（仅运行少数几条链）更短。为了确定由此产生的短链是否可靠，我们需要评估预热后马尔可夫链接近其平稳分布的程度。潜在尺度缩减因子 $\widehat R$ 是一种常用的收敛诊断工具，但不幸的是，它可能需要较长的采样阶段才能良好工作。我们提出了一种嵌套设计来克服这一挑战，并推广出一种称为嵌套 $\widehat R$ 的广义诊断方法。这种新诊断工具在与 $\widehat R$ 相似条件下有效，并完善了适用于GPU友好型采样器的工作流程。此外，所提出的嵌套结构为 $\widehat R$ 在经典和短链两种机制下的效用提供了理论见解。