On the toric algebra of graphical models

Details On the toric algebra of graphical models On the toric algebra of graphical models

2006-08-02

arxiv.org/abs/math/0608054v1

Annals of Statistics 2006, Vol. 34, No. 3, 1463-1492

Mathematics

Statistics Theory

Published at http://dx.doi.org/10.1214/009053606000000263 in the Annals of Statistics (http://www.imstat.org/aos/) by the Inst

Scientific paper

10.1214/009053606000000263

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.

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