ON FACTORIZATIONS OF PROBABILITY-DISTRIBUTIONS OVER DIRECTED-GRAPHS

Four notions of factorizability over arbitrary directed graphs are exa mined. For acyclic graphs they coincide and are identical with the usu al factorization of probability distributions in Markov models. Relati ons between the factorizations over circuits are described in detail i ncluding nontrivial counterexamples. Restrictions on the cardinality o f state spaces cause that a factorizability with respect to some speci al cyclic graphs implies the factorizability with respect to their, mo re simple, strict edge-subgraphs. This gives sometimes the possibility to break circuits and get back to the acyclic, well-understood case.