Reproducing Stein Kernel Approach for Post-Hoc Correction of Approximate Sampling Algorithms

We discuss a post-hoc rejection-free method of obtaining a consistent estimator for quantities related to a target distribution of interest by using samples obtained from an ergodic Markov chain with an arbitrary stationary distribution. The approach involves Stein importance sampling, based on minimization of the kernelized Stein discrepancy, and is shown to be valid under certain conditions on the mixing of the chain. To demonstrate the practical implications of the method, we show these conditions are satisfied for a large number of unadjusted samplers. We also conduct a numerical study showing that the method is super-efficient in practical scenarios that can involve both a large number of parameters and a large data set.