Conditional Independence with four random variables

Question

If A is conditionally independent of B and C given D. And that B and C are not conditionally independent given D. Can I write P(A,B,C|D) as P(A|D) * P(B,C|D)?

Thanks!

Answer 1

Yes! Although there is ambiguity what you mean, which AndréNicolas points out in the above comments, in either case the answer to your question is yes! The conditional independence of $A$ with $B$ and $C$ given $D$ (whether, you mean conditional independence with $B$ and conditional independence with $C$, or just conditional independence of $B \cap C$) is defined such that: $$P(A, B, C \mid D) = P(A \mid D) P(B,C \mid D)$$

Since $B$ and $C$ are not conditionally independent given $D$, you should also note that $P(B,C\mid D) \neq P(B\mid D) P(C \mid D)$.