# What does a semicolon denote in the context of probability and statistics?

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.

• 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. Commented Dec 9, 2017 at 21:23
• @XanderHenderson Mainly online notes, I think they are extremely prevalent. For example cs229.stanford.edu/notes/cs229-notes8.pdf Commented Dec 10, 2017 at 0:37

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.
• @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 $f_{\Psi_n|Y}(\psi_n|y;\lambda,\zeta)$..." Commented Dec 10, 2017 at 17:47