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. This characterization generalizes the well-known Hammersley-Clifford Theorem. We show that for decomposable graphical models these conditions are equivalent to a set of statistical independence facts as in the Hammersley-Clifford Theorem but that for non-decomposable graphical models they are not. We also show that non-decomposable models can have non-rational maximum likelihood estimates. Finally, using these results, we provide a characterization of decomposable graphical models.
[This is joint work with Dan Geiger and Bernd Sturmfels].