Stein's density method for multivariate continuous distributions

TL;DR

This paper develops a comprehensive framework for Stein's density method in multivariate continuous distributions, introducing new operators, Stein kernels, and bounds for distribution distances, with applications to normal, Student, and Bayesian posterior distributions.

Contribution

It introduces a unified approach to Stein's method for multivariate distributions, including new operators, Stein kernels, and explicit formulas, enhancing distribution comparison and approximation techniques.

Findings

Derived explicit formulas for Stein kernels of elliptical distributions.

Compared Stein discrepancies between various distributions, improving bounds in some cases.

Provided bounds on Wasserstein distances for multiple distribution comparisons, including Bayesian posteriors.

Abstract

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 $\mathbb{R}^d$, 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 Poincar\'e constant. We use these bounds to compare different copulas on the unit square in 1-Wasserstein distance.