# Measuring Sample Quality with Diffusions

arXiv: https://arxiv.org/abs/1611.06972 · HTML: https://mlsanity.com/paper/1611.06972/

## Key facts

- Cited by 25 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Nov 2016.

## Built on by (most cited first, 25 of 25)

- [Robust Generalised Bayesian Inference for Intractable Likelihoods](https://mlsanity.com/paper/2104.07359/) (2021)
- [A Survey of Monte Carlo Methods for Parameter Estimation](https://mlsanity.com/paper/2107.11820/) (2021)
- [Regularized Stein Variational Gradient Flow](https://mlsanity.com/paper/2211.07861/) (2022)
- [Sampling with Mirrored Stein Operators](https://mlsanity.com/paper/2106.12506/) (2021)
- [Targeted Separation and Convergence with Kernel Discrepancies](https://mlsanity.com/paper/2209.12835/) (2022)
- [Towards Healing the Blindness of Score Matching](https://mlsanity.com/paper/2209.07396/) (2022)
- [Controlling Moments with Kernel Stein Discrepancies](https://mlsanity.com/paper/2211.05408/) (2022)
- [Interpreting diffusion score matching using normalizing flow](https://mlsanity.com/paper/2107.10072/) (2021)
- [Efficient Sampling on Riemannian Manifolds via Langevin MCMC](https://mlsanity.com/paper/2402.10357/) (2024)
- [Bounding Wasserstein distance with couplings](https://mlsanity.com/paper/2112.03152/) (2021)
- [Geometric Methods for Sampling, Optimisation, Inference and Adaptive Agents](https://mlsanity.com/paper/2203.10592/) (2022)
- [Probabilistic Inference and Learning with Stein's Method](https://mlsanity.com/paper/2603.07467/) (2026)
- [A Unifying and Canonical Description of Measure-Preserving Diffusions](https://mlsanity.com/paper/2105.02845/) (2021)
- [Standardisation-function Kernel Stein Discrepancy: A Unifying View on Kernel Stein Discrepancy Tests for Goodness-of-fit](https://mlsanity.com/paper/2106.12105/) (2021)
- [Geometry-informed irreversible perturbations for accelerated convergence of Langevin dynamics](https://mlsanity.com/paper/2108.08247/) (2021)
- [Mean-Square Analysis of Discretized Ito Diffusions for Heavy-tailed Sampling](https://mlsanity.com/paper/2303.00570/) (2023)
- [Using Perturbation to Improve Goodness-of-Fit Tests based on Kernelized Stein Discrepancy](https://mlsanity.com/paper/2304.14762/) (2023)
- [De-randomizing MCMC dynamics with the diffusion Stein operator](https://mlsanity.com/paper/2110.03768/) (2021)
- [Bayesian Numerical Integration with Neural Networks](https://mlsanity.com/paper/2305.13248/) (2023)
- [Stein's method for marginals on large graphical models](https://mlsanity.com/paper/2410.11771/) (2024)
- [Fast and Scalable Score-Based Kernel Calibration Tests](https://mlsanity.com/paper/2510.14711/) (2025)
- [Efficient Learning of Stationary Diffusions with Stein-type Discrepancies](https://mlsanity.com/paper/2601.16597/) (2026)
- [Posterior sampling with CNN-based, Plug-and-Play regularization with applications to Post-Stack Seismic Inversion](https://mlsanity.com/paper/2212.14595/) (2022)
- [Correcting Mode Proportion Bias in Generalized Bayesian Inference via a Weighted Kernel Stein Discrepancy](https://mlsanity.com/paper/2503.02108/) (2025)
- [StAD: Stein Amortized Divergence for Fast Likelihoods with Diffusion and Flow](https://mlsanity.com/paper/2605.16486/) (2026)

## Abstract (arXiv)

Stein's method for measuring convergence to a continuous target distribution relies on an operator characterizing the target and Stein factor bounds on the solutions of an associated differential equation. While such operators and bounds are readily available for a diversity of univariate targets, few multivariate targets have been analyzed. We introduce a new class of characterizing operators based on Ito diffusions and develop explicit multivariate Stein factor bounds for any target with a fast-coupling Ito diffusion. As example applications, we develop computable and convergence-determining diffusion Stein discrepancies for log-concave, heavy-tailed, and multimodal targets and use these quality measures to select the hyperparameters of biased Markov chain Monte Carlo (MCMC) samplers, compare random and deterministic quadrature rules, and quantify bias-variance tradeoffs in approximate MCMC. Our results establish a near-linear relationship between diffusion Stein discrepancies and Wasserstein distances, improving upon past work even for strongly log-concave targets. The exposed relationship between Stein factors and Markov process coupling may be of independent interest.