On the Intersection Property of Conditional Independence and its Application to Causal Discovery

Abstract

This work investigates the intersection property of conditional independence. It states that for random variables A,B,C and X we have that X⊥⊥A|B,C and X⊥⊥B|A,C implies X⊥⊥(A,B)|C. Here, “ ⊥⊥” stands for statistical independence. Under the assumption that the joint distribution has a density that is continuous in A,B and C, we provide necessary and sufficient conditions under which the intersection property holds. The result has direct applications to causal inference: it leads to strictly weaker conditions under which the graphical structure becomes identifiable from the joint distribution of an additive noise model.