Stein's density approach and information inequalities

TL;DR

This paper extends Stein's density approach by introducing a new operator and class, enabling broader application to probability distributions and deriving improved information inequalities with explicit bounds.

Contribution

It introduces a new operator and class for Stein's density approach, leading to generalized information inequalities and improved bounds over existing results.

Findings

Derived a new Stein identity for broader distribution classes

Established explicit bounds for information inequality constants

Compared and improved upon Gaussian case inequalities

Abstract

We provide a new perspective on Stein's so-called density approach by introducing a new operator and characterizing class which are valid for a much wider family of probability distributions on the real line. We prove an elementary factorization property of this operator and propose a new Stein identity which we use to derive information inequalities in terms of what we call the \emph{generalized Fisher information distance}. We provide explicit bounds on the constants appearing in these inequalities for several important cases. We conclude with a comparison between our results and known results in the Gaussian case, hereby improving on several known inequalities from the literature.