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 with O(n^2) 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.