Support Sets in Exponential Families and Oriented Matroid Theory

TL;DR

This paper explores the relationship between support sets in exponential families and oriented matroid theory, providing a combinatorial framework to understand their closures and support structures.

Contribution

It introduces a novel connection between exponential family closures and oriented matroids, offering a combinatorial approach to analyze support sets.

Findings

Support sets in exponential family closures correspond to circuits of an oriented matroid.

Exponential families with the same induced oriented matroid have identical support set closures.

Positive cocircuits parameterize the closure of the exponential family.

Abstract

The closure of a discrete exponential family is described by a finite set of equations corresponding to the circuits of an underlying oriented matroid. These equations are similar to the equations used in algebraic statistics, although they need not be polynomial in the general case. This description allows for a combinatorial study of the possible support sets in the closure of an exponential family. If two exponential families induce the same oriented matroid, then their closures have the same support sets. Furthermore, the positive cocircuits give a parameterization of the closure of the exponential family.