Identifying the irreducible disjoint factors of a multivariate probability distribution

We study the problem of decomposing a multivariate probability distribution p(v) defined over a set of random variables V = {V1, . . . , Vn} into a product of factors defined over disjoint subsets {VF1 , . . . ,VFm}. We show that the decomposition of V into irreducible disjoint factors forms a unique partition, which corresponds to the connected components of a Bayesian or Markov network, given that it is faithful to p. Finally, we provide three generic procedures to identify these factors withO(n) pairwise conditional independence tests (Vi⊥ Vj |Z) under much less restrictive assumptions: 1) p supports the Intersection property; ii) p supports the Composition property; iii) no assumption at all.