Bayes Theorem chaining

Question

Is this statement true

$P(A|B)=P(A|C) \times P(C|B)$

If yes, how to prove it? If no, what should be minimal conditions on A, B and C for this to hold true?

Thanks for help.

Answer 1

I think the statement is false. A counter example would be welcome.

Here is a true statement. Let $C_i$ for $i=1..n$ a partition of $\Omega$, then we can write:

$P(A|B)=\frac{P(A,B)}{P(B)}=\sum_{i=1}^{n}{\frac{P(A,B,C_i)}{P(B)}}$

$\sum_{i=1}^{n}{P(A|B,C_i)\times P(C_i|B)} $

Answers, enlightenments are still very welcome.