Graph Relations

Let $I(\cdot)$ be the set of all conditional independencies consistent with the input graph/input distribution.

Independence Map (I-Map): Graph $G$ is an independence map for distribution $p$ if $I(G) \subseteq I(p)$. Intuitively, that means $p$ has at least the conditional independencies implied by the graph.

Perfect Map (P-Map): Graph $G$ is a perfect map for distribution $p$ if $I(G) = I(p)$. For a given undirected graph or a given undirected graph, there always exists some $p$ such that $G$ is a perfect map for $p$.

__Minimal I-Map$: Graph$G$is a minimal I-map for$p$if$G$is an I-map and removing any edges from the graph causes$G$$ to lose its status as an I-map.

To make these terms concrete, consider the following directed graph $G$:

\[X_1 \rightarrow X_3 \leftarrow X_2\]

Suppose $P(X_1, X_2, X_3) = P(X_1) P(X_2) P(X_3)$. Then $G$ is an I-map.
Suppose $$P(X_1, X_2, X_3) = P(X_1) P(X_2) P(X_3 X_1, X_2)$. Then$G$$ is a P-map.
Suppose $P(X_1, X_2, X_3) = P(X_2) P(X_1|X_2) P(X_3|X_1, X_2)$. Then $G$ is neither an I-map nor a P-map.