Given the following bayesian network:
I have to determine whether the following statement holds:
$P(B, H | E, G) = P(B | E, G) * P(H | E, G)$.
Is it true that this statement is valid when $B \bot H$? I don't really know which formula can be applied to a conditional probability of 2 joint probabilities, as is the case here.
The statement is true since $B$ and $H$ are conditionally independent given $G$, and $G$ is a given. The definition of independence is that $P(A, B) = P(A) * P(B)$
You could further prove this out by writing the probability for the whole system, summing over all the unwanted nodes, then saying that $P(H|G)$ is independent of its parents since $G$ is given. So $P(B, H|G) = P(B) * P(H|G)$.