Stein Π-Importance Sampling
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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 Π-invariant Markov chain to obtain a consistent approximation of P, the intended target. Surprisingly, the optimal choice of Π is not identical to the target P; we therefore propose an explicit construction for Π based on a novel variational argument. Explicit conditions for convergence of Stein Π-Importance Sampling are established. For ≈ 70% of tasks in the PosteriorDB benchmark, a significant improvement over the analogous post-processing of P-invariant Markov chains is reported.
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