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A distribution factorizes according to a Directed acyclic graph when its joint density is the product of the conditional density of each variable given its graphical parents. This factorization implies the graph's global Markov property of a directed acyclic graph.

A path in a causal directed acyclic graph is active given a conditioning set $K$ when every noncollider on the path is outside $K$ and every collider has itself or a descendant in $K$. Two vertex sets are d-separated by $K$ when no path between them is active given $K$. The global Markov property of a directed acyclic graph then turns d-separation into conditional independence.

In the displayed graph, the edges areEvery observed pair except $(X_1,X_4)$ and $(X_2,X_4)$ is joined by a direct edge, which remains active under conditioning on any other observed variables. The pair $(X_2,X_4)$ is always joined by the fork $X_2\leftarrow U\to X_4$, because the unobserved noncollider $U$ cannot be conditioned on.

For $(X_1,X_4)$, if $X_3$ is not conditioned on, $X_1\to X_3\to X_4$ is active. If $X_3$ is conditioned on, the pathbecomes active because its collider $X_2$ has conditioned descendant $X_3$; conditioning on $X_2$ itself also opens it. Thus every observed pair is d-connected given every subset of the other observed variables. Any conditional independence between two nonempty observed subvectors would imply one between each selected pair, so the graph entails none.