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 and X we have that X perpendicular to perpendicular to A vertical bar B; C and X perpendicular to perpendicular to B vertical bar A, C implies X perpendicular to perpendicular to (A, B) vertical bar C. Here, "perpendicular to perpendicular to" 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.