Induced Graph Semantics: Another look at the Hammersley-Clifford theorem
The Hammersley-Clifford (H-C) theorem relates the factorization properties of a probability distribution to the clique structure of an undirected graph. If a density factorizes according to the clique structure of an undirected graph, the theorem guarantees that the distribution satisfies the Markov property and vice versa. We show how to generalize the H-C theorem to different notions of decomposability and the corresponding generalized-Markov property. Finally we discuss how our technique might be used to arrive at other generalizations of the H-C theorem, inducing a graph semantics adapted to the modeling problem.