Stein discrepancies have emerged as a powerful tool for retrospective improvement of Markov chain Monte Carlo output. However, the question of how to design Markov chains that are well-suited to such post-processing has yet to be addressed. This paper studies Stein importance sampling, in which weights are assigned to the states visited by a $\Pi$-invariant Markov chain to obtain a consistent approximation of $P$, the intended target. Surprisingly, the optimal choice of $\Pi$ is not identical to the target $P$; we therefore propose an explicit construction for $\Pi$ based on a novel variational argument. Explicit conditions for convergence of Stein $\Pi$-Importance Sampling are established. For $\approx 70\%$ of tasks in the PosteriorDB benchmark, a significant improvement over the analogous post-processing of $P$-invariant Markov chains is reported.

翻译：斯坦因差异已成为一种强大的工具，用于对马尔可夫链蒙特卡洛输出进行回顾性改进。然而，如何设计适用于此类后处理的马尔可夫链这一问题尚未得到解决。本文研究斯坦因重要性抽样，其中为 $Π$ 不变的马尔可夫链所访问的状态分配权重，以获得对目标分布 $P$ 的一致近似。令人惊讶的是，$Π$ 的最优选择与目标 $P$ 并不相同；因此，我们基于一种新颖的变分论证提出了 $Π$ 的显式构造。建立了斯坦因 $Π$ 重要性抽样收敛的显式条件。在 PosteriorDB 基准测试中，约 70% 的任务报告了相对于 $P$ 不变马尔可夫链的类似后处理的显著改进。