Use of Bayes theorem in the Lovásk local lemma

Question

Here's a line from the proof on Wiki I don't understand.

$$\Pr(A\mid\bigwedge_{B\in S}\bar{B}) =\frac{\Pr(A\bigwedge_{B\in S_1}\bar{B} \mid \bigwedge_{B\in S_2}\bar{B})}{\Pr(\bigwedge_{B\in S_1}\bar{B} \mid \bigwedge_{B\in S_2} \bar{B})}.$$

We know that $A$ is independent from all the $B$s in $S_2$, and that $S$ is the union of $S_1$ and $S_2$.

The article just says that it follows from Bayes thereom, but I can't see that.

Answer 1

Let $C := \bigwedge_{B \in S_1}\bar{B}$ and $D := \bigwedge_{B \in S_2}\bar{B}$. The statement now reads like this:

$$\Pr(A\mid C \wedge D) =\frac{\Pr(A \wedge C \mid D)}{\Pr(C \mid D)}.$$

If $\text{Pr}_D(X)$ denotes $\text{Pr}(X \mid D)$, then the statement is equivalent to

$$\text{Pr}_D(A\mid C) =\frac{\text{Pr}_D(A \wedge C)}{\text{Pr}_D(C)},$$ which is Bayes' Theorem on $\text{Pr}_D$.