I am wondering what the parameter $y$ in the function $g(y,\mu,\sigma)=\frac{1}{(2\pi)^{1/2}\sigma}e^{-(y-\mu)^{2/2\sigma^2}}$ stands for in Section 6 (page 14) of the paper introducing the REINFORCE family of algorithms.
Drawing an analogy to Equation 4 of the same paper, I would guess that it refers to the outcome (i.e. sample) of sampling from a probability distribution parameterized by the parameters $\mu$ and $\sigma$. However, I am not sure whether that is correct or not.
If you take a look at the Wikipedia page related to the normal distribution, you will see the definition of the Gaussian density
$$ {\displaystyle f(x)={\frac {1}{\sigma {\sqrt {2\pi }}}}e^{-{\frac {1}{2}}\left({\frac {x-\mu }{\sigma }}\right)^{2}}} \label{1}\tag{1} $$
and you will see that the $y$ in your formula corresponds to the $x$ in equation \ref{1}.
I've seen this notation in the context of computer vision and image processing, where the Gaussian kernel is used to blur images.
So, as pointed out by someone in a comment, $y$ should indeed be the point where you evaluate the density.
Maybe the confusing part is that all parameters are treated equally in terms of their purpose, while $\mu$ and $\sigma$ are clearly the parameters that define the specific density, so they are not the inputs to the specific density.
After having read the relevant section of the paper, I now understand why you're confused. The author refers to $y$ as the output (not yet sure why: maybe it's the output of another unit that feeds this Gaussian unit?), but I think that this explanation still applies. The output of the Gaussian density $g$ is not $y$, but the density that corresponds to $y$. In fact, in appendix $B$ of the paper, the author says that $Y$ is the support of $g$ and $y$ is an element of $Y$.
It states that "To simplify notation, we focus on one single unit and omit the usual unit index subscript throughout"
So they are simply removing the i-th index from the equation for simplicity. So g is a function of a given instance "y" and the parameters μ and σ.
