Product of probabilities conditioning on different events

Question

Let $A,B,C,D$ be some events. Consider the probabilities $$ Pr(A|C) $$ and $$ Pr(B|D) $$ Under which conditions it holds that $$ Pr(A|C)\times Pr(B|D)=Pr(A,B|C,D) $$ ?

Answer 1

We, given the definition of conditional probability, we have that $$P(A|C)P(B|D) = \frac{P(A,C)}{P(C)}\cdot \frac{P(B,D)}{P(D)} = \frac{P(A,C)P(B,D)}{P(C)P(D)}.$$

Now similarly, $$P(A,B|C,D) = \frac{P(A,B,C,D)}{P(C,D)}.$$

Now, if $C$ and $D$ are independent, we have that $P(C,D) = P(C)P(D).$ Similarly, if $(A,C)$ is indendent of $(B,D)$, then $P(A,B,C,D)=P(A,C)P(B,D).$ Letting $\perp$ denote independence, we then have a sufficient (but not necessary) condition is $C \perp D, C \perp B, A \perp B, A\perp D.$