The <global Markov property for an undirected graph> says that whenever a vertex set $S$ separates disjoint sets $A$ and $B$ in the graph,
$$
V_A\mathrel{\perp\!\!\!\perp}V_B\mid V_S.
$$
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.

Solution

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

The global Markov property says that graph separation of vertex sets $A$ and $B$ by $S$ implies conditional independence of the corresponding random vectors given the variables at $S$.


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 Incoming links: Global Markov property for an undirected graph