D-separation

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In a bayesian network, an edge between X and Y means that there might be an unmediated dependence between these two variables, one that does not disappear no matter what information we receive from other variables. Conversely, the absence of an edge signifies the definite existence of a set of variables Z such that X and Y would be independent once we know the value of Z. For each pair $(X, Y)$ , the set Z that would render X independent of Y can be identified by a graphical condition called d-separation defined as follows.

A path p is said to be d-separated (or blocked) by a set of nodes Z if and only if

p contains a chain i -> m -> j or a fork i <- m -> j such that m is in Z, or
p contains an inverted fork (or collider) i -> m <- j such that neither m nor any descendant of m is in Z.

A set Z is said to d-separate X from Y in a directed acyclic graph if all paths from X to Y are d-separated by Z. The d in d-separation stands for "directional", since the behavior of a three-node link on a path in the definition depends on the direction of the arrows in the link.

An important result about bayesian networks is that whenever X and Y are d-separated by Z in a bayesian network, then X is independent of Y given Z in the joint probability distribution from which the bayesian network was generated.