notation: precedence of conditional when multiple variables

In expressions such as $P(X,Y|Z)$ and $I(X; Y|Z)$ (mutual information)

there are two interpretations for a student, and the correct one does not seem to be mentioned in textbooks.

1. "joint probability of (X and Y) given Z" and "(mutual information of X and Y) given Z"

2. "joint probability of X and (Y given Z)" and "mutual information of X and (Y given Z)"

For probability I think #1 is the correct one. But then, there could be an alternate notation that would be clearer, like $P(X|Z, Y|Z)$ and $I(X|Z; Y|Z)$.

I am afraid I have missed something very basic.

• Could you explain what (2) might actually mean? Perhaps you could give an example?
– whuber
Oct 2, 2017 at 15:46
• Yes, (2) makes no sense at all. Your answer is kind.
– Bull
Oct 9, 2017 at 13:21

• (+1) It is an unfortunate reality that many people think "$Y|Z$" is a random variable. Your answer is great by pointing out that "(it) only makes sense as $P(Y|Z)$" (although more precisely, you might mean "$P(Y \leq y | Z)$" or "$E(Y|Z)$"). Jan 22, 2023 at 19:41