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

This work investigates the intersection property of conditional independence. It states that for random variables A,B,C$$A,B,C$$ and X we have that X⊥⊥A|B,C$$X \bot \bot A{\kern 1pt} {\kern 1pt} |{\kern 1pt} {\kern 1pt} B,C$$ and X⊥⊥B|A,C$$X\, \bot \bot\, B{\kern 1pt} {\kern 1pt} |{\kern 1pt} {\kern 1pt} A,C$$ implies X⊥⊥(A,B)|C$$X\, \bot \bot\, (A,B){\kern 1pt} {\kern 1pt} |{\kern 1pt} {\kern 1pt} C$$. Here, “⊥⊥$$ \bot \bot $$” stands for statistical independence. Under the assumption that the joint distribution has a density that is continuous in A,B$$A,B$$ 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.