On the toric algebra of graphical models
Abstract
We formulate necessary and sufficient conditions for an arbitrary discrete probability distribution to factor according to an undirected graphical model, or a loglinear 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 HammersleyClifford 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.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 August 2006
 arXiv:
 arXiv:math/0608054
 Bibcode:
 2006math......8054G
 Keywords:

 Mathematics  Statistics;
 60E05;
 62H99 (Primary) 13P10;
 14M25;
 68W30 (Secondary)
 EPrint:
 Published at http://dx.doi.org/10.1214/009053606000000263 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)