Minimum relative entropy distributions with a large mean are Gaussian.

Entropy optimization principles are versatile tools with wide-ranging applications from statistical physics to engineering to ecology. Here we consider the following constrained problem: Given a prior probability distribution q, find the posterior distribution p minimizing the relative entropy (also known as the Kullback-Leibler divergence) with respect to q under the constraint that mean(p) is fixed and large. We show that solutions to this problem are approximately Gaussian. We discuss two applications of this result. In the context of dissipative dynamics, the equilibrium distribution of a Brownian particle confined in a strong external field is independent of the shape of the confining potential. We also 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 in time.