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

Specifically, these?

enter image description here

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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.

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    $\begingroup$ Nice succinct answer. Also want to add that it is generally read as "given", as in "the function of x given theta". $\endgroup$ – Underminer Jul 28 '20 at 15:52
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    $\begingroup$ 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". $\endgroup$ – Joe Jul 29 '20 at 1:02

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