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The global Markov property for an undirected graph says that whenever a vertex set $S$ separates disjoint sets $A$ and $B$ in the graph,The Hammersley-Clifford theorem states that a strictly positive density factorizes over the cliques of an undirected graph if and only if it satisfies this global Markov property. Strict positivity is essential for the converse from conditional independences to factorization.

Each factor $f_{j\mid \mathrm{pa}(j)}$ depends only on node $j$ and its parents. Those vertices form a clique in the moral graph, because moralization joins every pair of parents and removes arrow directions. The positive joint density therefore factors into clique potentials of the moral graph. The Hammersley-Clifford theorem then implies the global Markov property for that undirected graph.