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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.

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

1 Answer 1

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The problem with version 2., I think, is that (Y given Z) makes no sense. It is not a random variable itself, and only makes sense as P(Y|Z). The comment of whuber provided this answer to me.

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  • $\begingroup$ (+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 "$E(Y|Z)$"). $\endgroup$
    – Zhanxiong
    Jan 22, 2023 at 19:41

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