Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 221 / 2 / i

A positive density factorizes according to an undirected graphical model $G^u=(\mathcal{V},\mathcal{E})$ when there are nonnegative clique potentials ${\psi }_C$ such thatwhere $\mathcal{C}$ may be taken as the maximal cliques and $Z$ normalizes the density. Equivalently, each potential involves only variables in one complete subgraph.

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.