Sharp informational inequalities involving Kullback-Leibler and R\'enyi divergences and a family of scaling-invariant relative Fisher measures

TL;DR

This paper introduces a new transformation called relative differential-escort and a biparametric family of relative Fisher measures that are invariant under scaling, leading to sharp inequalities with Kullback-Leibler and Re9nyi divergences, and explores related measures and bounds.

Contribution

It proposes a novel relative differential-escort transformation and a family of scale-invariant relative Fisher measures with sharp inequalities involving divergences.

Findings

New relative Fisher measures are invariant under scaling.

Established sharp inequalities between the new measures and divergences.

Derived bounds and minimizing densities for the new measures.

Abstract

We introduce a new transformation called \emph{relative differential-escort}, which extends the usual differential-escort transformation by relating the change of variable to a reference probability density. As an application of it, we define a biparametric family of \emph{relative Fisher measures} presenting significant advantages with respect to the pre-existing ones in the literature: invariance under scaling changes and, consequently, sharp inequalities between the new relative Fisher measure and the well established Kullback-Leibler and R\'enyi divergences. We also introduce a biparametric family of \emph{relative cumulative moment-like measures} and we establish sharp lower bounds of these new measures by the Kullback-Leibler and R\'enyi divergences. The optimal bound and the minimizing densities are given. We also construct a family of inequalities for an arbitrary and fixed minimizing density in which the so-called generalized trigonometric functions plays a central role, providing thus one more interesting application of the newly introduced inequalities and measures.