Extreme points of sets of probability measures and $φ$-divergences

Abstract

In this work, we prove several equivalent characterizations of the extreme points of convex sets of probability measures of the form $\mathcal{M}=\mathcal{P} \cap H$, where $\mathcal{P}$ denotes the set of all probability measures on an arbitrary measurable space $(Ω,\mathcal{F})$ and $H$ is an affine set of signed measures on $(Ω,\mathcal{F})$ with finite variation. We first give a precise measure-theoretic formulation of the heuristic that extreme measures have minimal support. We then connect this with the notion of minimality with respect to absolute continuity, and prove that points that dominate no other element of $\mathcal{M}$ are the only ones realizing the blow-up of a certain divergence map for any suitable $ϕ$-divergence. Finally, considering a different class of $ϕ$-divergences, we recover a characterization of the extreme points of $\mathcal{M}$ as strict local maximizers of $ϕ$-divergences relative to any suitable dominating measures. We apply our result to recover and complement results from the literature in the context of finite spaces, sets of measures defined by integral constraints, multi-marginal couplings, and dominated sets of probability measures.