Stein operators, kernels and discrepancies for multivariate continuous distributions

Abstract

In this paper we present a general framework for Stein's method for multivariate continuous distributions. The approach gives a collection of Stein characterisations, among which we highlight score-Stein operators and kernel Stein operators. Applications include copulas and distance between posterior distributions. We give a general construction for Stein kernels for elliptical distributions and discuss Stein kernels in generality, highlighting connections with Fisher information and mass transport. Finally, a goodness-of-fit test based on Stein discrepancies is given.

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• Rate of Convergence of Polynomial Networks to Gaussian Processes
• A Fourier representation of kernel Stein discrepancy with application to Goodness-of-Fit tests for measures on infinite dimensional Hilbert spaces
• Stein-type covariance identities: Klaassen, Papathanasiou and Olkin-Shepp type bounds for arbitrary target distributions
• Stability estimates for invariant measures of diffusion processes, with applications to stability of moment measures and Stein kernels
• Stein's Method for Probability Distributions on ${\mathbb{S}}^1$