Graphoids

In the Directed Acyclic Graph (DAG) setting, we say a DAG \(G=([p],E)\) encodes the conditional independence model \(J(G) = \{I(A,B|C) : A,B \text{ are d-separated by } C \text{ in } G\}\).

Given a set of random variables \(X_1,\cdots,X_p\), with distribution \(\mathbb{P}\) we say a collection of conditional independence statements \(J(\mathbb{P})\) is a graphoid if they satisfy the graphoid axioms, given any subsets \(X,Y,Z,W\) with empty intersection and denoting \(I(A,B|C)\) are \(A,B\) being conditionally independent given \(C\):

Intuitive explanations for the graphoid axioms are stated in (, , p. 12). Namely:

For a generalization of the graphoid axioms to the context-specific case where the conditional independences hold given a certain event (e.g. \(Z=k\) instead of just \(Z\)), refer to (, a, Section 3).

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