Support sets in exponential families and oriented matroid theory

Authors:
Johannes Rauh;Thomas Kahle;Nihat Ay
Affiliations:
Max Planck Institute for Mathematics in the Sciences, Inselstr. 22, D-04103 Leipzig, Germany;Max Planck Institute for Mathematics in the Sciences, Inselstr. 22, D-04103 Leipzig, Germany and Institut Mittag-Leffler, Auravägen 17, 182 60 Djursholm, Sweden;Max Planck Institute for Mathematics in the Sciences, Inselstr. 22, D-04103 Leipzig, Germany and Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA
Venue:
International Journal of Approximate Reasoning
Year:
2011

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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.