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This paper provides a general framework for Stein's density method formultivariate continuous distributions. The approach associates to anyprobability density function a canonical operator and Stein class, as well asan infinite collection of operators and classes which we call standardizations.These in turn spawn an entire family of Stein identities and characterizationsfor any continuous distribution on $\mathbb{R}^d$, among which we highlightthose based on the score function and the Stein kernel. A feature of theseoperators is that they do not depend on normalizing constants. A new definitionof Stein kernel is introduced and examined; integral formulas are obtainedthrough a connection with mass transport, as well as ready-to-use explicitformulas for elliptical distributions. The flexibility of the kernels is usedto compare in Stein discrepancy (and therefore 2-Wasserstein distance) betweentwo normal distributions, Student and normal distributions, as well as twonormal-gamma distributions. Upper and lower bounds on the 1-Wassersteindistance between continuous distributions are provided, and computed for avariety of examples: comparison between different normal distributions(improving on existing bounds in some regimes), posterior distributions withdifferent priors in a Bayesian setting (including logistic regression), centredAzzalini--Dalla Valle distributions. Finally the notion of weak Stein equationand weak Stein factors is introduced. Bounds for solutions of the weak Steinequation are obtained for Lipschitz test functions if the distribution admits aPoincar\'e constant. We use these bounds to compare different copulas on theunit square in 1-Wasserstein distance.

Stein's density method for multivariate continuous distributions