Recent developments in Markov chain Monte Carlo (MCMC) algorithms allow us to run thousands of chains in parallel 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 \textit{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：马尔可夫链蒙特卡洛（MCMC）算法的最新进展允许我们利用GPU等硬件加速器，以近乎运行单链的速度并行执行数千条链。尽管每条链仍需在热身阶段忘记其初始点，但随后的采样阶段可以比传统设置（仅运行少量链）更短。为了确定这些短链的可靠性，我们需要评估热身结束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指标，但遗憾的是，它可能需要较长的采样阶段才能有效工作。我们提出了一种嵌套设计来应对这一挑战，并推广出一种名为\textit{嵌套} $\widehat R$ 的新诊断方法。该新诊断方法在类似 $\widehat R$ 的条件下工作，并完善了适用于GPU友好型采样器的流程。此外，所提出的嵌套设计为 $\widehat R$ 在经典和短链场景中的效用提供了理论见解。