Stein $\Pi$-Importance Sampling
Main Conference Track
Authors: Congye Wang, Ye Chen, Heishiro Kanagawa, Chris J. Oates

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.