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A density factorizes according to a Directed acyclic graph whenwhere $\mathrm{pa}(j)$ is the set of parents of vertex $j$.

The required undirected graph is the moral graph: join every pair of parents having a common child, retain the parent-child adjacencies, and remove all arrowheads. Each DAG family $\{j\}\cup \mathrm{pa}(j)$ is then a clique, so each conditional factor is a clique potential and the DAG factorization is also an undirected factorization. These edges are minimal for a guarantee covering every DAG-factorizing density, because an arbitrary conditional factor can couple every pair of variables in its family.

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.