Is it acceptable/usual to write multiple conditional probability pipes simultaneously? eg, $\tilde P(L \mid M_2)\equiv P(L \mid M_2\mid M_1)$

Solution 1:

The notation with two vertical lines is wrong, and it comes down to what precisely the vertical line means in conditional probability.

$P(A\mid B)$ is "the probability of $A$, assuming $B$ is known to be true", not the probability of some event $A\mid B$. In that sense, $P(A\mid B)$ and $P(A)$ are quite different functions, despite having the same name $P$.

We are given the definition that $\tilde P(A)=P(A\mid M_1)$. This definition doesn't actually tell us what $\tilde P(A\mid B)$ should be, but we can infer that it should be something like "the probability of $A$, assuming $B$ is known to be true, assuming $M_1$ is known to be true."

This, as was written in the linked paper, means that $\tilde P(L\mid M_2)$ equivalent to $P(L\mid M_1\cap M_2)$.