  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.
