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June 2006 On the toric algebra of graphical models
Dan Geiger, Christopher Meek, Bernd Sturmfels
Ann. Statist. 34(3): 1463-1492 (June 2006). DOI: 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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Dan Geiger. Christopher Meek. Bernd Sturmfels. "On the toric algebra of graphical models." Ann. Statist. 34 (3) 1463 - 1492, June 2006.


Published: June 2006
First available in Project Euclid: 10 July 2006

zbMATH: 1104.60007
MathSciNet: MR2278364
Digital Object Identifier: 10.1214/009053606000000263

Primary: 60E05 , 62H99
Secondary: 13P10 , 14M25 , 68W30

Keywords: Conditional independence , decomposable models , factorization , factorization of discrete distributions , graphical models , Gröbner bases , Hammersley–Clifford theorem

Rights: Copyright © 2006 Institute of Mathematical Statistics


Vol.34 • No. 3 • June 2006
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