TITLE: If $A$ and $B$ are independent, what can we say about $P(X\mid A,B)$?
QUESTION [5 upvotes]: Given two independent events A and B, what can we say about $P(X\mid A,B)$ ?
Is the following correct ?
$$P(X\mid A,B) = \frac{P(A,B\mid X)P(X)}{P(A,B)}\tag{Bayes}$$
$$P(X\mid A,B) = \frac{P(A\mid X)P(B\mid X)P(X)}{P(A)P(B)} \tag{Independence}$$
$$P(X\mid A,B) = \frac{\frac{P(X\mid A)P(A)}{P(X)}\frac{P(X\mid B)P(B)}{P(X)}P(X)}{P(A)P(B)} \tag{Bayes}$$
$$P(X\mid A,B) = \frac{P(X\mid A)P(X\mid B)}{P(X)}$$

REPLY [8 votes]: Is the following correct ?

No, the "independence" step is bogus since it requires that A and B are independent conditionally on X, a condition which is not implied by the independence of A and B.