A study of some entropy inequalities
Date
2023
Authors
Journal Title
Journal ISSN
Volume Title
Publisher
University of Delaware
Abstract
In this document we investigate some entropy inequalities. We divide these into two parts. ☐ In the first part we prove a new class of inequalities for submodular set functions, indexed by chordal graphs. Since entropy is a particularly useful example of a submodular function, we deduce some entropy inequalities. As a further corollary, we construct a novel family of determinant inequalities for sums of positive definite Hermitian matrices, and also recover an inequality of Barrett, Johnson, and Lundquist (1989). ☐ In the second part we introduce a new inequality called entropic symmetrization resistance. An asymmetric random variable X is said to be variance (respectively, en- tropic) symmetrization-resistant if every independent random variable Y that produces a symmetric sum X+Y has a greater variance (respectively, entropy) than that of X. Asymmetric Bernoulli random variables were shown to be variance symmetrization- resistant by Kagan, Mallows, Shepp, Vanderbei, and Vardi (1999); Pal (2008) gave a proof using stochastic calculus. We give a third proof. We show for the first time that asymmetric Bernoulli random variables are entropic symmetrization resistant. We then extend the entropy and variance inequalities to the hypercube and explore other possible extensions to non-Bernoulli random variables.
Description
Keywords
Entropy inequalities, Entropic symmetrization, Hermitian matrices, Random variables, Variance inequalities