Identifying the irreducible disjoint factors of a multivariate probability distribution

We study the problem of decomposing a multivariate probability distribution p(\mathbfv) defined over a set of random variables \mathbfV={V_1,…,V_n} into a product of factors defined over disjoint subsets {\mathbfV_F_1,…,\mathbfV_F_m}. We show that the decomposition of \mathbfV 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 (V_i\perp V_j \mathbin∣\mathbfZ) under much less restrictive assumptions: 1) p supports the Intersection property ii) p supports the Composition property iii) no assumption at all.