A concavity property for the reciprocal of Fisher information and its consequences on Costa's EPI

TL;DR

This paper demonstrates that for log-concave densities, the reciprocal of Fisher information is concave when Gaussian noise is added, leading to a strengthened form of Costa's entropy power inequality with implications for information theory.

Contribution

It establishes the concavity of the reciprocal of Fisher information for log-concave densities under Gaussian noise addition, extending Costa's EPI.

Findings

Reciprocal of Fisher information is concave in t for log-concave densities.

Third derivative of entropy power is nonnegative for log-concave densities.

Improves upon Costa's entropy power inequality for log-concave distributions.

Abstract

We prove that the reciprocal of Fisher information of a log-concave probability density $X$ in ${\bf{R}}^n$ is concave in $t$ with respect to the addition of a Gaussian noise $Z_t = N(0, tI_n)$. As a byproduct of this result we show that the third derivative of the entropy power of a log-concave probability density $X$ in ${\bf{R}}^n$ is nonnegative in $t$ with respect to the addition of a Gaussian noise $Z_t$. For log-concave densities this improves the well-known Costa's concavity property of the entropy power.