The reproducing Stein kernel approach for post-hoc corrected sampling
This provides a theoretical foundation for post-hoc correction of approximate sampling algorithms, which is incremental but extends applicability to Markov chains and general spaces.
The paper tackles the problem of correcting biased samples from Markov chains using Stein importance sampling, proving that this approach yields consistent estimators for target distributions even when the chain's invariant measure differs from the target. The results apply to general Polish spaces and retain consistency with data subsampling.
Stein importance sampling is a widely applicable technique based on kernelized Stein discrepancy, which corrects the output of approximate sampling algorithms by reweighting the empirical distribution of the samples. A general analysis of this technique is conducted for the previously unconsidered setting where samples are obtained via the simulation of a Markov chain, and applies to an arbitrary underlying Polish space. We prove that Stein importance sampling yields consistent estimators for quantities related to a target distribution of interest by using samples obtained from a geometrically ergodic Markov chain with a possibly unknown invariant measure that differs from the desired target. The approach is shown to be valid under conditions that are satisfied for a large number of unadjusted samplers, and is capable of retaining consistency when data subsampling is used. Along the way, a universal theory of reproducing Stein kernels is established, which enables the construction of kernelized Stein discrepancy on general Polish spaces, and provides sufficient conditions for kernels to be convergence-determining on such spaces. These results are of independent interest for the development of future methodology based on kernelized Stein discrepancies.