On the Geometry of Mixtures of Prescribed Distributions

Abstract

We consider the space of w-mixtures that are finite statistical mixtures sharing the same prescribed component distributions, like Gaussian mixture models sharing the same components. The information geometry induced by the Kullback-Leibler (KL) divergence yields a dually flat space where the KL divergence between two w-mixtures amounts to a Bregman divergence for the negative Shannon entropy generator, called the Shannon information. Furthermore, we prove that the skew Jensen-Shannon statistical divergence between w-mixtures amount to skew Jensen divergences on their parameters and state several divergence inequalities between w-mixtures and their closures.