Is $P(A,B|C)$ the same as $P(A|B)P(B|C)$?

Question

I am trying to derive Markov property and encounter $P(A,B|C)$, can I do this $P(A,B|C)=P(A|B)P(B|C)$? This can give me a desired result, but I doubt if this is legal.

Answer 1

In general, $P(A,B \mid C) = P(A\mid \color{red}{B,C}) P(B\mid C)$.

Intuitively, if you want to know whether $A$ and $B$ both happen, given that $C$ happened, then you:

• First, check if $B$ happened; the answer is yes with probability $P(B \mid C)$, since we already know $C$ happened.
• If yes, then second, check if $A$ happened; the answer is yes with probability $P(A \mid B,C)$, since we already know $B$ and $C$ happened.

Your identity does not hold unless $P(A \mid B,C) = P(A\mid B)$.