Notation problem for Gaussian Distribution. Vertical Bar

Question

In "Gaussian Process Latent Variable Models forVisualisation of High Dimensional Data" by Lawrence I stepped over the following definitions of Gaussian Distributions:

$p(x) = N(x|0,I)$

$p(y|x,W,\beta) = N(y|Wx,\beta^{-1}I)$

$\rightarrow$ Full definition

What is the meaning of the vertical Bar in $N(\cdot|\cdot,\cdot)$? And what are the actuall mean $\mu$ of the distrubutions?

Answer 1

In general, $f(x|a, b)$ means you should consider $f$ as a function of $x$, and think of $a$ and $b$ as the parameters of the function. Here, $N(x|0,I)$ means probability density of $x$ according to the normal distribution with mean $0$ (vector zero) and variance $I$. You can think of mean and the variance as the parameters of the probability density function. Also, the conditional probability density of $y$ is determined by a normal distribution with mean $Wx$ and variance $\beta^{-1} I$.