A stroll along the gamma

TL;DR

This paper introduces a detailed study of the 'smart path' interpolation between arbitrary probability measures and gamma distributions, providing new formulas, identities, and inequalities that extend classical results.

Contribution

It presents novel explicit representations, a new notion of relative Fisher information, and extends classical identities and inequalities to the gamma distribution context.

Findings

New explicit representation formulas for the smart path.

Differential and integrated De Bruijn identities for gamma distributions.

A new proof of the gamma logarithmic Sobolev inequality and a novel HSI inequality.

Abstract

We provide the first in-depth study of the "smart path" interpolation between an arbitrary probability measure and the gamma-$(\alpha, \lambda)$ distribution. We propose new explicit representation formulae for the ensuing process as well as a new notion of relative Fisher information with a gamma target distribution. We use these results to prove a differential and an integrated De Bruijn identity which hold under minimal conditions, hereby extending the classical formulae which follow from Bakry, Emery and Ledoux's $\Gamma$-calculus. Exploiting a specific representation of the "smart path", we obtain a new proof of the logarithmic Sobolev inequality for the gamma law with $\alpha\geq 1/2$ as well as a new type of HSI inequality linking relative entropy, Stein discrepancy and standardized Fisher information for the gamma law with $\alpha\geq 1/2$.