# Meaning of vertical bar | in loss function?

Does anyone know what the vertical bars in these equations here mean?

Specifically, these? It denotes that the function is parameterized by $$\theta$$ and the $$x_i$$ are the inputs to the function. For example $$f(x|\theta)=x\cdot \theta$$ is the dot product of the input, $$x$$, and the parameters $$\theta$$. User @Underminer adds a note about reading: if you wanted to read the symbols $$f(x|\theta)$$ aloud, you might say "the function $$f$$ of $$x$$ given $$\theta$$."
Some other usages are described at Wikipedia's article https://en.wikipedia.org/wiki/Vertical_bar#Mathematics. Incidentally, this was my first Google hit for the search math vertical bar notation.
• Sometimes this is also read as "f of x conditional on $\theta$". In traditional statistics sometimes there is another notation $f(x;\theta)$ which is always "f of x given $\theta$", as $\theta$ is not a random variable, but in Bayesian statistics, and sometimes machine learning, it is considered a random variable, hence "conditional upon" or "conditioned on". – Joe Jul 29 '20 at 1:02