Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 221 / 3 / d / Solution

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.