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

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.

Let $K={\Sigma }^{-1}$ be the precision matrix, let $R=\{1,\dots ,p\}\setminus \{j,k\}$, and condition on $V_R=v_R$. In the Gaussian exponent, all terms depending jointly on $v_j$ and $v_k$ are contained inIf $K_{jk}=0$, this conditional density is a product of one function of $v_j$ and one function of $v_k$, so the conditional independence of $V_j$ and $V_k$ given $V_R$ holds.

Conversely, conditional independence makes this everywhere-positive conditional density factorize. Its mixed second derivative must therefore vanish:Hence the conditional independence holds exactly when $({\Sigma }^{-1}{)}_{jk}=0$.

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.