Open Access
2023 Stein’s density method for multivariate continuous distributions
G. Mijoule, M. Raič, G. Reinert, Y. Swan
Author Affiliations +
Electron. J. Probab. 28: 1-40 (2023). DOI: 10.1214/22-EJP883


This paper provides a general framework for Stein’s density method for multivariate continuous distributions. The approach associates to any probability density function a canonical operator and Stein class, as well as an infinite collection of operators and classes which we call standardizations. These in turn spawn an entire family of Stein identities and characterizations for any continuous distribution on Rd, among which we highlight those based on the score function and the Stein kernel. A feature of these operators is that they do not depend on normalizing constants. A new definition of Stein kernel is introduced and examined; integral formulas are obtained through a connection with mass transport, as well as ready-to-use explicit formulas for elliptical distributions. The flexibility of the kernels is used to compare in Stein discrepancy (and therefore 2-Wasserstein distance) between two normal distributions, Student and normal distributions, as well as two normal-gamma distributions. Upper and lower bounds on the 1-Wasserstein distance between continuous distributions are provided, and computed for a variety of examples: comparison between different normal distributions (improving on existing bounds in some regimes), posterior distributions with different priors in a Bayesian setting (including logistic regression), centred Azzalini–Dalla Valle distributions. Finally the notion of weak Stein equation and weak Stein factors is introduced. Bounds for solutions of the weak Stein equation are obtained for Lipschitz test functions if the distribution admits a Poincaré constant. We use these bounds to compare different copulas on the unit square in 1-Wasserstein distance.

Funding Statement

GM gratefully acknowledges support by the Fonds de la Recherche Scientifique – FNRS under Grant MIS F.4539.16. YS acknowledges support by the Fonds de la Recherche Scientifique – FNRS under Grant CDR/OL J.0197.20, as well as the ULB ARC Consolidator grant. GR acknowledges support from EPSRC grants EP/T018445/1, EP/R018472/1, and EP/X0021951, and the Alan Turing Institute.


We are indebted to Andreas Kyprianou for his patient and considerate handling of this paper. MR would like to thank Oliver Dragičević for his help with Sobolev spaces. We also thank Max Fathi, Christophe Ley, Gilles Mordant and Guillaume Poly for interesting discussions, as well as Lester Mackey and Steven Vanduffel for suggesting some references which we had overlooked.


Download Citation

G. Mijoule. M. Raič. G. Reinert. Y. Swan. "Stein’s density method for multivariate continuous distributions." Electron. J. Probab. 28 1 - 40, 2023.


Received: 13 November 2019; Accepted: 10 November 2022; Published: 2023
First available in Project Euclid: 2 May 2023

arXiv: 2101.05079
MathSciNet: MR4583066
Digital Object Identifier: 10.1214/22-EJP883

Primary: 60B10 , 60B12

Keywords: density method , elliptical distributions , Stein kernels , Stein operators

Vol.28 • 2023
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