update: I find the post on Double conditional probability @Joe referenced to be useful. But I am still trying to figure out some details.
conditional probability states $$f(A) = \int f(A|z)f(z)dz$$ What happens when $f(A)$ is also a conditional probability? If $f(A)=f(X|Y)$, then what is $f(X|Y|Z)$?
Speaking the math in English, $f(X|Y|Z)$ seems to be the probability of $X$ given $Y$ given $Z$. Since $Y$ and $Z$ are given, it seems like saying the probability of $X$ given $Y$ and $Z$, which is $f(X|Y\cap Z)$. But this conclusion feels weird.
To sum up, what is $f(X|Y|Z)$, and how do you represent it with the p.d.f's of $X$, $Y$, $Z$ or their conditional probabilities? (with at most one $|$ notation, since I am having trouble understanding two |s)
