Distances between distributions via Stein's method

TL;DR

This paper advances Stein's method by developing new solution representations and bounds for Stein equations, enabling improved bounds on distribution distances like Kolmogorov, Total Variation, and Wasserstein, with competitive results across various examples.

Contribution

It introduces novel representations and bounds for solutions to Stein equations, enhancing the ability to measure distances between distributions.

Findings

New uniform and non-uniform bounds on Stein solutions.

Effective bounds on distribution distances including Wasserstein and Total Variation.

Results are competitive with existing literature across multiple examples.

Abstract

We build on the formalism developed in [arXiv:1906.08372v1] to propose new representations of solutions to Stein equations. We provide new uniform and non uniform bounds on these solutions (a.k.a.\ Stein factors). We use these representations to obtain representations for differences between expectations in terms of solutions to the Stein equations. We apply these to compute abstract Stein-type bounds on Kolmogorov, Total Variation and Wasserstein distances between arbitrary distributions. We apply our results to several illustrative examples, and compare our results with current literature on the same topic, whenever possible. In all occurrences our results are competitive.