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

Abstract

We prove that the reciprocal of Fisher information of a log-concave probability density $X$ in ${\mathbf{R}}^n$ is concave in $t$ with respect to the addition of a Gaussian noise $Z_t=N(0,t{I}_n)$. As a byproduct of this result we show that the third derivative of the entropy power of a log-concave probability density $X$ in ${\mathbf{R}}^n$ 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.

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