Support Sets in Exponential Families and Oriented Matroid Theory

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.

27 pages, extended version published in IJAR

References in corpus (3)

• On the toric algebra of graphical models
• Closures of exponential families
• Computing generating sets of lattice ideals

Cited by in corpus (6)

• Expressive Power and Approximation Errors of Restricted Boltzmann Machines
• Exponential Varieties
• Finding the Maximizers of the Information Divergence from an Exponential Family
• A note on the border of an exponential family
• Approximating faces of marginal polytopes in discrete hierarchical models
• On the closure of relational models