On the Toric Algebra of Graphical Models

Dan Geiger, Christopher Meek, and Bernd Sturmfels

Abstract

We formulate necessary and sufficient conditions for an arbitrary discrete probability distribution to factor according to an undirected graphical model, or a log-linear model, or other more general exponential models. For decomposable graphical models these conditions are equivalent to a set of conditional independence statements similar to the Hammersley–Clifford theorem; however, we show that for nondecomposable graphical models they are not. We also show that nondecomposable models can have nonrational maximum likelihood estimates. These results are used to give several novel characterizations of decomposable graphical models.

Details

Publication typeArticle
Published inThe Annals of Statistics
URLhttp://arxiv.org/abs/math/0608054
Pages1463-1492
Volume34
Number3
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