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

This work investigates the intersection property of conditional independence.\nIt states that for random variables $A,B,C$ and $X$ we have that $X$\nindependent of $A$ given $B,C$ and $X$ independent of $B$ given $A,C$ implies\n$X$ independent of $(A,B)$ given $C$. Under the assumption that the joint\ndistribution has a continuous density, we provide necessary and sufficient\nconditions under which the intersection property holds. The result has direct\napplications to causal inference: it leads to strictly weaker conditions under\nwhich the graphical structure becomes identifiable from the joint distribution\nof an additive noise model.\n