  This work investigates the intersection property of conditional independence.
It states that for random variables $A,B,C$ and $X$ we have that $X$
independent of $A$ given $B,C$ and $X$ independent of $B$ given $A,C$ implies
$X$ independent of $(A,B)$ given $C$. Under the assumption that the joint
distribution has a continuous density, 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.
