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I see this kind of notation often

$$ p_{\theta} (x|z, y) = f(x; z, y, \theta) $$

I understand the conditional prob noation on the left. What is the significance of the ; on the joint prob on the right?

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    $\begingroup$ My preference is to write $X|Y, Z; \theta, \phi$, to show conditional distributions, conditional on other random variables, and fixed, unknown parameters. $\endgroup$ – tchakravarty Sep 4 '17 at 19:54
  • $\begingroup$ You've got different guesses. Perhaps you could give one or two examples from published work and more context. $\endgroup$ – Nick Cox Sep 5 '17 at 6:36
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What this notation says is that

$$ f(x; z, y, \theta) $$

is a function of $x$ with "parameters" $z, y, \theta$. It is just a way to visually show that they are of different kind (e.g. data vs parameters, random vs fixed quantities). While the conditional notation has precise meaning, this notation is used informally, to improve readability of the formulas rather than to convey any specific meaning.

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  • $\begingroup$ $z, y$ would be unusual notation for parameters. With no other information and if obliged to guess I would go for @tchakrcarty's interpretation. $\endgroup$ – Nick Cox Sep 5 '17 at 6:35
  • $\begingroup$ Sorry; that should be @tchakravarty. $\endgroup$ – Nick Cox Sep 5 '17 at 7:37

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