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)?
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up.Sign up to join this community
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 $P(B,C\mid D) \neq P(B\mid D) P(C \mid D)$.