Support Sets in Exponential Families and Oriented Matroid Theory

We discuss how to obtain an implicit description of the closure of a discrete exponential family with a finite set of equations derived from 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 framework allows us to study the possible support sets of an exponential families with the help of oriented matroid theory. In particular, if two exponential families induce the same oriented matroid, then they have the same support sets.