The Annals of Statistics

On the toric algebra of graphical models

Dan Geiger, Christopher Meek, and Bernd Sturmfels

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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.

Article information

Ann. Statist. Volume 34, Number 3 (2006), 1463-1492.

First available: 10 July 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60E05: Distributions: general theory 62H99: None of the above, but in this section
Secondary: 13P10: Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) 14M25: Toric varieties, Newton polyhedra [See also 52B20] 68W30: Symbolic computation and algebraic computation [See also 11Yxx, 12Y05, 13Pxx, 14Qxx, 16Z05, 17-08, 33F10]

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


Geiger, Dan; Meek, Christopher; Sturmfels, Bernd. On the toric algebra of graphical models. The Annals of Statistics 34 (2006), no. 3, 1463--1492. doi:10.1214/009053606000000263.

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