Stein $\Pi$-Importance Sampling
Congye Wang, Wilson Ye Chen, Heishiro Kanagawa, Chris J. Oates
Abstract
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.
BibTeX
@inproceedings{
wang2023stein,
title={Stein \${\textbackslash}Pi\$-Importance Sampling},
author={Congye Wang and Wilson Ye Chen and Heishiro Kanagawa and Chris J. Oates},
booktitle={Thirty-seventh Conference on Neural Information Processing Systems},
year={2023},
url={https://openreview.net/forum?id=wiidCRA3at}
}