If I have a Bayesian network of the form:
$A\rightarrow B\rightarrow C\rightarrow D$
is it necessarily true that $P(B,D|C)=P(B|C)P(D|C)$?
Wikipedia gives the following defining condition for a Bayes net: 'variable is conditionally independent of its non-descendants given its parent variables', where non-descendant means no direct arrow connecting the random variables.
On the left-hand side of the above equation, one conditions on the only parent of $D$, which is $C$. So $D$ should be conditionally independent of $B$, even though no conditioning on the parents of $B$ takes places, yes?