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 $$X \bot \bot A{\kern 1pt} {\kern 1pt} |{\kern 1pt} {\kern 1pt} B,C$$ and $$X\, \bot \bot\, B{\kern 1pt} {\kern 1pt} |{\kern 1pt} {\kern 1pt} A,C$$ implies $$X\, \bot \bot\, (A,B){\kern 1pt} {\kern 1pt} |{\kern 1pt} {\kern 1pt} C$$ . Here, “ $$ \bot \bot $$ ” stands for statistical independence. Under the assumption that the joint distribution has a density that is continuous in $$A,B$$ 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.