Stein \Pi -Importance Sampling

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.