Consider the following paragraph from this research paper:
To learn the generator’s distribution $p_g$ over data $x$, we define a prior on input noise variables $p_z(z)$, then represent a mapping to data space as $G(z;θ_g)$, where $G$ is a differentiable function represented by a multilayer perceptron with parameters $θ_g$. We also define a second multilayer perceptron $D(x;θ_d)$ that outputs a single scalar. $D(x)$ represents the probability that $x$ came from the data rather than $p_g$.
Keep the parameters aside for this discussion and consider only domain and range.
distribution $p_g$ over data $x$
I am thinking that the term distribution refers to probability density function $p_g$ over a tuple $X = (X_1, X_2, X_3, \cdots, X_n)$ of random variables. That means $p_g$ takes a particular assignment of random variables and gives a real value that signifies the density of probability at that point. So, $$p_g : X \rightarrow R$$
Assume that the domain $X$ is a set of all possible assignments of random variables.
prior on input noise variables $p_z(z)$
I am thinking that $p_z$ is also a probability density function from $R^n \rightarrow R$.
represent a mapping to data space as $G(z;θ_g)$
I am interpreting the dataspace as the set of all possible assignments of random variables i.e., $X$. So, $G$ takes $R^n$ as domain and $X$ as range.
$D(x;θ_d)$ that outputs a single scalar. $D(x)$ represents the probability
I am interpreting $D$ as a probability measure that takes $R^n$ as input (may be from dataset $X$ or random from $R^n$) and outputs an element from $[0,1]$.
Are my interpretations for 1, 2, 3 and 4 correct in case of continuous random variables?
Note: I am using $X$ as tuple $(X_1, \cdots, X_n)$ as well as Cartesian product of $X_1,X_2, \cdots X_n$ i.e., $X = X_1 \times X_2 \times \cdots \times X_n$ for notational convenience.