# Why are A->C<-B conditionally dependent in a directed graph?

$P(A,B,C) = P(A)P(B)P(C|A,B)$. I understand how $A,B$ are marginally independent on $C$, but I'm confused as to how the $A, B$ are conditionally dependent on $C$.

$P(A,B|C) = \frac{P(A)P(B)P(C|A,B)}{P(C)}$, and $P(C)$ is the marginal of $C$ over $P(A)P(B)P(C|A,B)$, but that was as far as I got.

If they would be conditionally independent, it would be possible to decompose $$P(A,B|C)=P(A|C)P(B|C)\textrm{.}$$ However, as you correctly write $$P(A,B|C) = \frac{P(A)P(B)P(C|A,B)}{P(C)}\textrm{.}$$ Thus, whether $A$ and $B$ are conditionally dependent depends on the decomposition of $P(C|A,B)=Q_1(A,C)Q_2(B,C)$ where $Q_1$ and $Q_2$ are some nonnegative functions. If they exist, $A$ and $B$ are conditionally independent for given $C$.