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I sometimes see the notation $;$ being used in a statistical context

For example, let $f_X(x)$ be the probability density function associated with random variable $X$, then sometimes I see things like $f_X(x| y; \theta)$, where $\theta$ is a set of the mean and the covariance associated with the distribution.

What does these $;$ mean? Any reference helps.

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  • $\begingroup$ Where are you seeing this notation? Can you provide a link or reference to the place where it has been written? Cross Validated says that it divides data from parameters. $\endgroup$
    – Xander Henderson ♦
    Commented Dec 9, 2017 at 21:23
  • $\begingroup$ @XanderHenderson Mainly online notes, I think they are extremely prevalent. For example cs229.stanford.edu/notes/cs229-notes8.pdf $\endgroup$
    – Fraïssé
    Commented Dec 10, 2017 at 0:37

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It's used as a grand kind of comma. In this case, I think $x$ and $y$ refer to possible values of random variables, and $\theta$ an unknown non-random parameter, and the conditional density of $x$ given $y$ might be denoted $f(x|y)$, but the author wants to emphasize the fact that the whole expression depends on $\theta$ as well, and so might have written $f(x|y,\theta).$ But because in his world $y$ and $\theta$ are different kinds of things, writes $f(x|y;\theta)$ instead.

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  • $\begingroup$ Do you have a reference for this? $\endgroup$
    – Xander Henderson ♦
    Commented Dec 9, 2017 at 21:22
  • $\begingroup$ No. I'm away from my library just now, but am sure if I flipped through enough maths stats books, I'd see it, as I have seen it in the past. Maybe without an explicit discussion of the notation, but in a context where the meaning is clear. If I find an example, I'll post it. $\endgroup$
    – kimchi lover
    Commented Dec 9, 2017 at 21:35
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    $\begingroup$ @XanderHenderson My copy of Theoretical Statistics (1974 edition) by Cox and Hinkley, p.401, eqn (59) has an instance of this notation. It says: "Now the posterior density of $\Psi_n$ for fixed $\lambda,\zeta$ and data $y$ involves the data only through $y_n$ and is $f_{\Psi_n|Y}(\psi_n|y;\lambda,\zeta)$..." $\endgroup$
    – kimchi lover
    Commented Dec 10, 2017 at 17:47
  • $\begingroup$ I think a more accurate discussion of this notation can be found here: stats.stackexchange.com/questions/10234/… $\endgroup$
    – aaronsnoswell
    Commented Mar 20, 2020 at 9:22
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    $\begingroup$ Hi @kimchilover, My understanding is that the semicolon actually carries mathematical meaning, rather than just being a stylistic preference. The discussion I linked highlights that the semicolon indicates the objects to the right of the ';' are parameters, i.e. fixed inputs to the distribution, while the use of a bar indicates the objects to the right of '|' are random variables. So calling them fancy commas actually confuses the meaning. Also, it's not my tribe, I'm just a confused engineer trying to understand some notation I came across :) $\endgroup$
    – aaronsnoswell
    Commented Mar 27, 2020 at 2:15

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