Stein’s method

The importance of possessive apostrophes

A famous generic method for approximating distributions and quantifying discrepancy and manufacturing concentration bounds and limit theorems is Stein’s method, typically in the form of the method of exchangeable pairs (Stein 1986, 1972). Wikipedia will do as a rough intro for now, although their info is rather out-of-date. There does not seem to be a thorough introduction to all the modern and useful tools here. Different bits are introduced in Barbour and Chen (2005); Chatterjee (2014); Meckes (2012); Ross (2011).

Chen Soon Ong says:

If we have a distribution P, and we want to measure the distance to P from another distribution Q (which we control), an interesting trick to measure this distance is to define an operator T. This operator T, called the Stein operator, allows us to measure the distance between distributions by considering the distance between test functions on the random variables corresponding to P and Q. This is a more general structure than integral probability metrics, which in turn is a more general version of Wasserstein distance.

Probably the best intro is Lily Li’s Whirlwind Tour.

Stein operators


The original form, Stein’s lemma (Stein 1972) gives use the Stein operator for the Gaussian distribution in particular. Meckes (2009) explains:

The normal distribution is the unique probability measure \(\mu\) for which \[ \int\left[f^{\prime}(x)-x f(x)\right] \mu(d x)=0 \] for all \(f\) for which the left-hand side exists and is finite. It is useful to think of this in terms of operators, specifically, the operator \(\mathcal{A}_{o}\) defined on \(C^{1}\) functions by \[ \mathcal{A}_{o} f(x)=f^{\prime}(x)-x f(x) \] is called the characterizing operator of the standard normal distribution.
This is incredibly useful in probability approximation by Gaussians where it justifies Stein’s method, below. It has apparently been extended to elliptical distributions and exponential families.

Multivariate? Why, yes please. The following lemma of Meckes (2006) gives a second-order characterizing operator for the Gaussian distribution on \(\mathbb{R}^{k}\):

For \(f \in C^{1}\left(\mathbb{R}^{k}\right)\), define the gradient of \(f\) by \(\nabla f(x)=\left(\frac{\partial f}{\partial x_{1}}(x), \ldots, \frac{\partial f}{\partial x_{k}}(x)\right)^{t}\). Define the Laplacian of \(f\) by \(\Delta f(x)=\sum_{i=1}^{k} \frac{\partial^{2} f}{\partial x_{i}^{2}}(x)\). Now, let \(Z \sim \mathcal{N}(0_k, \mathrm{I}_k)\).

  1. If \(f: \mathbb{R}^{k} \rightarrow \mathbb{R}\) is two times continuously differentiable and compactly supported, then \[ \mathbb{E}[\Delta f(Z)-Z \cdot \nabla f(Z)]=0 \]
  2. If \(Y \in \mathbb{R}^{k}\) is a random vector such that \[ \mathbb{E}[\Delta f(Y)-Y \cdot \nabla f(Y)]=0 \] for every \(f \in C^{2}\left(\mathbb{R}^{k}\right)\), then \(\mathcal{L}(Y)=\mathcal{L}(Z) .\)
  3. If \(g \in C_{o}^{\infty}\left(\mathbb{R}^{k}\right)\), then the function \[ U_{o} g(x):=\int_{0}^{1} \frac{1}{2 t}[\mathbb{E} g(\sqrt{t} x+\sqrt{1-t} Z)-\mathbb{E} g(Z)] d t \] is a solution to the differential equation \[ \Delta h(x)-x \cdot \nabla h(x)=g(x)-\mathbb{E} g(Z) \]


a.k.a. Stein-Chen. \[\mathcal{A}_{o} f(k)=\lambda f(k+1)-k f(k)\]


Ross (2011) mentions

\[\mathcal{A}_{o} f(x)=f^{\prime}(x)-f(x)+f(0)\]

Markov processes

TBD; relation to infinitesimal generators? See perhaps Schoutens (2001).

Face to face, each classic case
We shadow box and double cross
Yet need the chase
A license to love, insurance to hold
Melts all your memories and change into gold
His eyes are like angels but his heart is cold
No need to ask
He’s a Stein operator
Stein operator
Stein operator
Stein operator

(With apologies to Sade.)

Stein’s method of exchangeable pairs

Meckes (2009) summarises.

Heuristically, the univariate method of exchangeable pairs goes as follows. Let \(W\) be a random variable conjectured to be approximately Gaussian; assume that \(\mathbb{E} W=0\) and \(\mathbb{E} W^{2}=1 .\) From \(W,\) construct a new random variable \(W^{\prime}\) such that the pair \(\left(W, W^{\prime}\right)\) has the same distribution as \(\left(W^{\prime}, W\right) .\) This is usually done by making a “small random change” in \(W\), so that \(W\) and \(W^{\prime}\) are close. Let \(\Delta=W^{\prime}-W\). If it can be verified that there is a \(\lambda>0\) such that \[ \begin{aligned} \mathbb{E}[\Delta \mid W]=-\lambda W+E_{1} \\ \mathbb{E}\left[\Delta^{2} \mid W\right]=2 \lambda+E_{2} \\ \mathbb{E}|\Delta|^{3}=E_{3} \end{aligned} \] with the random quantities \(E_{1}, E_{2}\) and the deterministic quantity \(E_{3}\) being small compared to \(\lambda,\) then \(W\) is indeed approximately Gaussian, and its distance to Gaussian (in some metric) can be bounded in terms of the \(E_{i}\) and \(\lambda\).

This comes out very nicely where there are natural symmetries to exploit, e.g. in low-d projections.

Non-Gaussian Stein method

Steins method generalises to AFAICT any exponential distribution. TBD

Multivariate Gaussian Stein method

The work of Elizabeth Meckes (1980—2020) serves as the canonical introduction in the area for now, although she never wrote a textbook. Two foundational ones are Chatterjee and Meckes (2008) and Meckes (2009) and there is a kind of introductory user guide in Meckes (2012); The examples are mostly about random projections although the method is much more general. The exchangeable pairs are natural in projections though, you can just switch off your brain and turn the handle to produce results, or easier yet, a computer algebra system that can handle noncommutative algebra can do it for you.

If the papers are too dense, try this friendly lecture, Stein’s Method — The last gadget under the hood.

Stein discrepancy

A probability metric based on something like “how well this distribution satisfies Stein’s lemma” I think?

I would like to read the Kernelized Stein gradient descent tutorial.


Agueh, Martial, and Guillaume Carlier. 2011. “Barycenters in the Wasserstein Space.” SIAM Journal on Mathematical Analysis 43 (2): 904–24.
Anastasiou, Andreas, Alessandro Barp, François-Xavier Briol, Bruno Ebner, Robert E. Gaunt, Fatemeh Ghaderinezhad, Jackson Gorham, et al. 2021. “Stein’s Method Meets Statistics: A Review of Some Recent Developments.” arXiv:2105.03481 [math, Stat], May.
Arras, Benjamin, and Christian Houdré. 2019a. On Stein’s Method for Infinitely Divisible Laws with Finite First Moment. Edited by Benjamin Arras and Christian Houdré. SpringerBriefs in Probability and Mathematical Statistics. Cham: Springer International Publishing.
———. 2019b. “On Stein’s Method for Multivariate Self-Decomposable Laws.” Electronic Journal of Probability 24 (none): 1–63.
Barbour, A. D., and Louis H. Y. Chen, eds. 2005. An Introduction to Stein’s Method. Vol. 4. Lecture Notes Series / Institute for Mathematical Sciences, National University of Singapore, v. 4. Singapore : Hackensack, N.J: Singapore University Press ; World Scientific.
Chatterjee, Sourav. 2014. “A Short Survey of Stein’s Method.” arXiv:1404.1392 [math], April.
Chatterjee, Sourav, and Elizabeth Meckes. 2008. “Multivariate Normal Approximation Using Exchangeable Pairs.” arXiv:math/0701464, January.
Chen, Louis H. Y. 1998. “Stein’s Method: Some Perspectives with Applications.” In Probability Towards 2000, edited by L. Accardi and C. C. Heyde, 97–122. Lecture Notes in Statistics. New York, NY: Springer.
Chwialkowski, Kacper, Heiko Strathmann, and Arthur Gretton. 2016. “A Kernel Test of Goodness of Fit.” In Proceedings of the 33rd International Conference on International Conference on Machine Learning - Volume 48, 2606–15. ICML’16. New York, NY, USA:
Cuturi, Marco, and Arnaud Doucet. 2014. “Fast Computation of Wasserstein Barycenters.” In International Conference on Machine Learning, 685–93. PMLR.
Gorham, Jackson, and Lester Mackey. 2015. “Measuring Sample Quality with Stein’s Method.” In Advances in Neural Information Processing Systems. Vol. 28.
Gorham, Jackson, Anant Raj, and Lester Mackey. 2020. “Stochastic Stein Discrepancies.” arXiv:2007.02857 [cs, Math, Stat], October.
Landsman, Zinoviy, and Johanna Nešlehová. 2008. “Stein’s Lemma for Elliptical Random Vectors.” Journal of Multivariate Analysis 99 (5): 912–27.
Landsman, Zinoviy, Steven Vanduffel, and Jing Yao. 2013. “A Note on Stein’s Lemma for Multivariate Elliptical Distributions.” Journal of Statistical Planning and Inference 143 (11): 2016–22.
Ley, Christophe, Gesine Reinert, and Yvik Swan. 2017. “Stein’s Method for Comparison of Univariate Distributions.” Probability Surveys 14 (none): 1–52.
Liu, Qiang, Jason D. Lee, and Michael I. Jordan. 2016. “A Kernelized Stein Discrepancy for Goodness-of-Fit Tests and Model Evaluation.” arXiv:1602.03253 [stat], July.
Liu, Qiang, and Dilin Wang. 2019. “Stein Variational Gradient Descent: A General Purpose Bayesian Inference Algorithm.” In Advances In Neural Information Processing Systems.
Loh, Wei-Liem. n.d. “Stein’s Method and Multinomial Approximation.”
Meckes, Elizabeth. 2006. “An Infinitesimal Version of Stein’s Method of Exchangeable Pairs.”
———. 2009. “On Stein’s Method for Multivariate Normal Approximation.” In High Dimensional Probability V: The Luminy Volume, 153–78. Beachwood, Ohio, USA: Institute of Mathematical Statistics.
———. 2012. “Approximation of Projections of Random Vectors.” Journal of Theoretical Probability 25 (2): 333–52.
Paulin, Daniel, Lester Mackey, and Joel A. Tropp. 2016. “Efron–Stein Inequalities for Random Matrices.” The Annals of Probability 44 (5): 3431–73.
Piccoli, Benedetto, and Francesco Rossi. 2016. “On Properties of the Generalized Wasserstein Distance.” Archive for Rational Mechanics and Analysis 222 (3): 1339–65.
Reinert, Gesine. 2000. “Stein’s Method for Epidemic Processes.” In Complex Stochastic Systems, 235–75. Boca Raton: Chapman & Hall/CRC.
———. 2005. “Three General Approaches to Stein’s Method.” In An Introduction to Stein’s Method, Volume 4:183–221. Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore, Volume 4. CO-PUBLISHED WITH SINGAPORE UNIVERSITY PRESS.
Reinert, Gesine, and Adrian Röllin. 2007. “Multivariate Normal Approximation with Stein’s Method of Exchangeable Pairs Under a General Linearity Condition,” November.
Ross, Nathan. 2011. “Fundamentals of Stein’s Method.” Probability Surveys 8 (0): 210–93.
Schoutens, Wim. 2001. “Orthogonal Polynomials in Stein’s Method.” Journal of Mathematical Analysis and Applications 253 (2): 515–31.
Stein, Charles. 1972. “A Bound for the Error in the Normal Approximation to the Distribution of a Sum of Dependent Random Variables.” Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Volume 2: Probability Theory, January, 583–602.
———. 1986. Approximate Computation of Expectations. Vol. 7. IMS.
Tan, Yue, Yingdong Lu, and Cathy Xia. 2018. “Relative Error of Scaled Poisson Approximation via Stein’s Method.” arXiv:1810.04300 [math], October.
Upadhye, Neelesh S., and Kalyan Barman. 2020. “A Unified Approach to Stein’s Method for Stable Distributions.” arXiv:2004.07593 [math], July.
Xu, Wenkai, and Takeru Matsuda. 2020. “A Stein Goodness-of-Fit Test for Directional Distributions.” In International Conference on Artificial Intelligence and Statistics, 320–30. PMLR.
———. 2021. “Interpretable Stein Goodness-of-Fit Tests on Riemannian Manifolds.” arXiv:2103.00895 [stat], March.
Xu, Wenkai, and Gesine Reinert. 2021. “A Stein Goodness of Fit Test for Exponential Random Graph Models.” arXiv:2103.00580 [stat], February.

No comments yet. Why not leave one?

GitHub-flavored Markdown & a sane subset of HTML is supported.