Collection Nexus Trimester - 2016 - Fundamental Inequalities and Lower Bounds Theme
Organizer(s)
Date(s)
12/05/2024
Beyond Shannon EPI: a Completely Monotone Conjecture
By Fan Cheng
Let XX be an arbitrary random variable and ZZ be an independent Gaussian random $~N(0,1)N(0,1)$. The differential entropy $h(X+(\sqrt t)Z), t>0$ plays a fundamental role in information theory. In Costa's entropy power inequality, $e^{2h}(X+t\sqrt Z)$ is showed to be concave in $t$. Noting that Fisher information $I(X+t\sqrt Z)$ is the first derivative of $h(X+t\sqrt Z)$. In our work, we introduce a conjecture on the signs of all the derivatives of $I(X+t\sqrt Z): (−1)^n \partial^n/\partial t^n I(X+t\sqrt Z)\geq 0$, for $n=0,1,2,…$ That is, $I(X+t\sqrt Z)$ is completely monotone in $t$. The conjecture may date back to a problem studied by McKean in mathematical physics in 1966, which remained unknown to information theory until very recently. In this talk, we will introduce the background, the progress, and the implication of the completely monotone conjecture.
