Finite undirected graphs, Markov properties, and clique factorization

Assume AC. Let $G=(V,E)$ be a finite simple undirected graph, with $E$ a set of unordered two-element subsets of $V$. A clique is a nonempty subset $C\subseteq V$ whose distinct vertices are adjacent. Let $(X_i{)}_{i\in V}$ be random elements on a common probability space, and for $U\subseteq V$ put ${\mathcal{F}}_U=\sigma (X_i:i\in U)$, with ${\mathcal{F}}_{\mathrm{\varnothing }}=\{\mathrm{\varnothing },\mathrm{\Omega }\}$. A path is a finite sequence of distinct vertices with adjacent consecutive vertices; $C$ separates $A$ from $B$ if every path from $A$ to $B$ meets $C$. The pairwise Markov property is ${\mathcal{F}}_{\{i\}}\perp \text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\perp {\mathcal{F}}_{\{j\}}\mid {\mathcal{F}}_{V\setminus \{i,j\}}$ whenever $i\mathrm{\ne }j$ and $\{i,j\}\mathrm{\notin }E$. The global Markov property is ${\mathcal{F}}_A\perp \text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\perp {\mathcal{F}}_B\mid {\mathcal{F}}_C$ whenever $A,B,C\subseteq V$ are pairwise disjoint and $C$ separates $A$ from $B$. If each $X_i$ takes values in a finite nonempty state space ${\mathcal{X}}_i$, a finite-state clique-factorized mass function on $G$ is $p(x)=Z^{-1}{\prod }_{C\in \mathcal{C}}{\psi }_C(x_C)$, where $\mathcal{C}$ is a finite collection of nonempty cliques, ${\psi }_C:{\prod }_{i\in C}{\mathcal{X}}_i\to (0,\mathrm{\infty })$, and $Z={\sum }_{x\in {\prod }_{i\in V}{\mathcal{X}}_i}{\prod }_{C\in \mathcal{C}}{\psi }_C(x_C)$. The product state space and sum are finite, and every potential is positive and finite, so $0<Z<\mathrm{\infty }$ and $p$ is a probability mass function. No equivalence between the clique-factorized class and either Markov property is asserted here.

The Elements of Statistical Learning, §17.2

Evans, Graphical Models, §§4.1–4.3