Stein Thinning


Optimally thinning of output from a sampling procedure, such as MCMC. Here the red samples are automatically chosen by Stein Thinning to provide a more accurate approximation to the distributional target, compared with the original MCMC output. [Read more]

View the Project on GitHub wilson-ye-chen/stein_thinning_start


Stein Thinning is a tool for post-processing the output of a sampling procedure, such as Markov chain Monte Carlo (MCMC). It aims to minimise a Stein discrepancy, selecting a subsequence of samples that best represent the distributional target.

The user provides two arrays: one containing the samples and another containing the corresponding gradients of the log-target. Stein Thinning returns a vector of indices, indicating which samples were selected.

In favourable circumstances, Stein Thinning is able to:


Implementations of Stein Thinning are available for Python, R, and MATLAB:

Get Started

First, it is important to parametrise the distributional target so that it has a positive and differentiable density formula on formula.

In Python, R, or MATLAB, it takes a single function call to start Stein Thinning:

indices = thin(samples, gradients, m)


Stan Examples

Stein Thinning can be used to post-process the output directly from the Stan family of probabilistic programming languages:

Citing Stein Thinning

Riabiz M, Chen WY, Cockayne J, Swietach P, Niederer SA, Mackey L, Oates CJ (2022) Optimal Thinning of MCMC Output. Journal of the Royal Statistical Society, Series B, 84(4), 1059-1081. arXiv

Papers Inspired By Stein Thinning

Benard C, Staber B, Da Veiga S (2023) Kernel Stein Discrepancy thinning: a theoretical perspective of pathologies and a practical fix with regularization. arXiv

Fisher MA, Oates CJ (2022) Gradient-Free Kernel Stein Discrepancy arXiv

Anastasiou A, Barp A, Briol F-X, Ebner B, Gaunt RE, Ghaderinezhad F, Gorham J, Gretton A, Ley C, Liu Q, Mackey L, Oates CJ, Reinert G, Swan Y (2023) Stein’s Method Meets Computational Statistics: A Review of Some Recent Developments. Statistical Science, 38(1), 120–139. arXiv

Hawkins C, Koppel A, Zhang Z (2022) Online, Informative MCMC Thinning with Kernelized Stein Discrepancy. arXiv

Teymur O, Gorham J, Riabiz M, Oates CJ (2021) Optimal Quantisation of Probability Measures Using Maximum Mean Discrepancy. International Conference on Artificial Intelligence and Statistics (AISTATS 2021). Conference arXiv

Chopin N, Ducrocq G (2021) Fast compression of MCMC output. Entropy, 23(8), 1017. Journal arXiv

South LF, Riabiz M, Teymur O, Oates C (2021) Post-Processing of MCMC. Annual Reviews of Statistics and its Application, 9, 529-555. arXiv


This work was supported by the Lloyd’s Register Foundation Programme on Data-Centric Engineering and the Programme on Health and Medical Sciences at The Alan Turing Institute (UK), the British Heart Foundation (SP/18/6/33805, RG/15/9/31534, PG/15/91/31812, FS/18/27/33543), the UK Research and Innovation Strategic Priorities Fund (EP/T001569/1), the UK Engineering and Physical Sciences Research Council (EP/P01268X/1, NS/A000049/1, EP/M012492/1), the UK National Institute for Health Research (II-LB-1116-20001) and the Wellcome Trust (WT 203148/Z/16/Z).