This question is really a comment on: what is the semicolon notation in joint probability?. I attached a short version of this comment on it -- written as a rhetorical question -- but then was asked to post this as a separate question. So here is the comment as question.
If a pdf $f_{X|Y,Z}(X|Y,Z; \theta)$ is proper, then the normalization, moments, etc. perforce depend on $Y=y, Z=z$ and $\theta.$ That is, the semicolon has no meaning whatsoever for the operational use of the pdf.
Then what meaning could the semicolon have?
It seems to me that it could only indicate that the author reserves the right to use the rules of joint and conditional probability to write $f_{Y|X, Z}(Y|X, Z;\theta)$ or $f_{X, Y|Z}(X, Y|Z; \theta)$, etc., but promises never to write $f_{\theta| X, Y, Z}(\theta |X, Y, Z)$ or $f_{\theta, Z|X, Y}(\theta, Z |X, Y)$ or some such. The semicolon is an abstract way of saying "I promise not to open the question of the random variable $\Theta$ in the present context." Why such a promise is ever necessary is not clear to me.
My question: do I have the sense of the semicolon notation right?
From a Bayesian point of view, both parameters and data are routinely represented as random variables, so the oft-cited semicolon-as-parameter-vs.-data-distinguisher is not so much a logical statement as revelation of the author's intentions and/or biases. It is subtle and elegant, perhaps, but it is also a distraction, and it raises more questions than it settles. I can't imagine a scenario when using the semicolon would be better than eschewing it in favor of an ordinary comma.
Addendum: perhaps the semicolon is an indicator that a posterior for $\theta$ could only ever be improper. (That's just a conjecture.)
