# The reproducing Stein kernel approach for post-hoc corrected sampling

arXiv: https://arxiv.org/abs/2001.09266 · HTML: https://mlsanity.com/paper/2001.09266/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Jan 2020.

## Built on by (most cited first, 10 of 10)

- [Kernel Stein Discrepancy Descent](https://mlsanity.com/paper/2105.09994/) (2021)
- [Gradient Estimation with Discrete Stein Operators](https://mlsanity.com/paper/2202.09497/) (2022)
- [Targeted Separation and Convergence with Kernel Discrepancies](https://mlsanity.com/paper/2209.12835/) (2022)
- [A kernel Stein test of goodness of fit for sequential models](https://mlsanity.com/paper/2210.10741/) (2022)
- [Geometric Methods for Sampling, Optimisation, Inference and Adaptive Agents](https://mlsanity.com/paper/2203.10592/) (2022)
- [Using Perturbation to Improve Goodness-of-Fit Tests based on Kernelized Stein Discrepancy](https://mlsanity.com/paper/2304.14762/) (2023)
- [The Minimax Lower Bound of Kernel Stein Discrepancy Estimation](https://mlsanity.com/paper/2510.15058/) (2025)
- [Minimax Lower Bounds of Kernel Discrepancy Estimation: MMD, HSIC, KSD](https://mlsanity.com/paper/2607.24235/) (2026)
- [Scalable Monte Carlo for Bayesian Learning](https://mlsanity.com/paper/2407.12751/) (2024)
- [Copula Discrepancy: Benchmarking Dependence Structure](https://mlsanity.com/paper/2507.21434/) (2025)

## Abstract (arXiv)

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.