Is this form of bayes rule valid?

Question

In the bayes rule, does \begin{equation} \Pr((A\mid B)\mid C) \end{equation}

have a meaning? is it a valid form of a probability? if $(A\mid B)$ ( I do not mean $\Pr(A\mid B)$ ) is an event, the aforementioned form of bayes rule would be legal. If yes, then in

\begin{equation} \Pr((A\mid B)\mid C)=\frac{\Pr((A\mid B),C)}{\Pr(C)} \end{equation}

is there any trick to calculate $\Pr((A\mid B),c)$?

P.S: $A,B,C$ are dependent to each other.

Answer 1

$(A\mid C)$ is not an event. It is a simple misunderstanding to think that $\Pr(A\mid C)$ is the probability of something called $A\mid C$ or $(A\mid C)$ or $A$ given $C$. The expression $\Pr(A\mid C)$ is not the probability of something called $A$ given $C$. Rather, it is the probability given $C$, of $A$. Grammatically, one should not read it as being the thing you get when you put $A\mid C$ into $\Pr(\cdots)$, but rather as the thing you get when you put $A$ into $\Pr(\cdots\mid C)$.