Minimum relative entropy distributions with a large mean are Gaussian

TL;DR

This paper demonstrates that for a large mean constraint, the distribution minimizing relative entropy relative to a prior tends to be Gaussian, with implications for evolutionary dynamics and entropy increase.

Contribution

It establishes the asymptotic Gaussianity of minimum relative entropy distributions with large means and applies this to evolutionary entropy dynamics.

Findings

Solutions are asymptotically Gaussian for large mean constraints.

Entropy of fitness distribution increases over time under natural selection.

Provides a new perspective on constrained distribution optimization.

Abstract

We consider the following frustrated optimization problem: given a prior probability distribution $q$, find the distribution $p$ minimizing the relative entropy with respect to $q$ such that $\textrm{mean}(p)$ is fixed and large. We show that solutions to this problem are asymptotically Gaussian. As an application we derive an $H$-type theorem for evolutionary dynamics: the entropy of the (standardized) distribution of fitness of a population evolving under natural selection is eventually increasing.