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