The Annals of Statistics
- Ann. Statist.
- Volume 34, Number 3 (2006), 1463-1492.
On the toric algebra of graphical models
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.
Ann. Statist. Volume 34, Number 3 (2006), 1463-1492.
First available in Project Euclid: 10 July 2006
Permanent link to this document
Digital Object Identifier
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]
Geiger, Dan; Meek, Christopher; Sturmfels, Bernd. On the toric algebra of graphical models. Ann. Statist. 34 (2006), no. 3, 1463--1492. doi:10.1214/009053606000000263. https://projecteuclid.org/euclid.aos/1152540755