Entropy-variance inequalities for discrete log-concave random variables via degree of freedom

TL;DR

This paper establishes a sharp inequality relating min-entropy and variance for integer-valued log-concave variables, identifying the geometric distribution as the minimizer, and extends to a discrete entropy power inequality.

Contribution

It introduces a discrete degree of freedom concept to prove a new min-entropy-variance inequality for log-concave variables, improving existing entropy power inequalities.

Findings

Geometric distribution minimizes min-entropy for fixed variance among log-concave sequences.

Derived a discrete Rényi entropy power inequality for log-concave variables.

Enhanced previous results by Bobkov, Marsiglietti, and Melbourne (2022).

Abstract

We utilize a discrete version of the notion of degree of freedom to prove a sharp min-entropy-variance inequality for integer valued log-concave random variables. More specifically, we show that the geometric distribution minimizes the min-entropy within the class of log-concave probability sequences with fixed variance. As an application, we obtain a discrete R\'enyi entropy power inequality in the log-concave case, which improves a result of Bobkov, Marsiglietti and Melbourne (2022).