# Meaning of the notation $\mathbb{E}_{x\sim \mu_{\text{ref}}, y\sim \mu_D(x)}[\ln y]$ in GAN framework

A very naive question about notations used in mathematical framework for generative adversarial network (GAN). What is precise mathematical definition of terms like

$$\mathbb{E}_{x\sim \mu_{\text{ref}}, y\sim \mu_D(x)}[\ln y]$$

as appearing in wiki intro.

In generally, if $$X: \Omega \to \mathbb{R}$$ is a random variable on probability space $$(\Omega, P)$$ with density function $$f_X$$ and $$g: \mathbb{R} \to \mathbb{R}$$, then $$\mathbb{E}[g(X)]= \int g(X) dP_X= \int g(x) f_X(x)dx$$ by definition.

My question is what means purely mathematically this expected value looking like object $$\mathbb{E}_{x\sim \mu_{\text{ref}}, y\sim \mu_D(x)}[\ln y]$$ with the bottom notation $$x\sim \mu_{\text{ref}}, y\sim \mu_D(x)$$ in the expression above? Is this bottom notation just an additional reminder that the expected value is formed as before $$\mathbb{E}[\ln y]$$ with respect to two random variables $$x$$ and $$y$$, where the former has density function $$\mu_{\text{ref}}$$ and the latter $$\mu_D(x)$$? Right, of does it mean something different?

Note, that $$\mu_D$$ is valued in probability measures, therefore $$\mathbb{E}_{x\sim \mu_{\text{ref}}, y\sim \mu_D(x)}[\ln y]$$ should probably better considered as "parameterized family of distributions". But I do not understand how this object looks to be explicitly written out?

Maybe the question is trivial, but in classical stochastics I'm a little bit familiar with I never came across such bottom notation for expected values, so seems to be nonstandard "GAN specific" terminology, therefore I would like to clarify it's precise mathematical meaning.

Specifically in this part of GAN's loss function, you are essentially considering an expectation over the conditional distribution of assessment samples $$\mu_D(x)$$ as they vary with different $$x$$ drawn from the reference distribution $$\mu_{\text{ref}}$$. Thus you're right that it should be considered as an expectation over a parameterized family of distributions and it's a way to express the expectation over both the randomness in choosing $$x$$ from the real data distribution and the randomness in generating $$y$$ from the discriminator network. The GAN training process involves adjusting the parameters of $$\mu_D(x)$$ to minimize or maximize its entire loss function aimed by the generator and the discriminator, respectively. ​
• ok so just to sum up: For any fix random variable $X$ with distribution function $f_X$ the expectation with bottom notation $\mathbb{E}_{X \sim f_X}(g(X)]$ and the "standard" notation for expectation $\mathbb{E}(g(X)]$ (by def $:= \int g(x)f_X(x)dx$) are the same, right? And specifically in the context on GAN formalism $\mathbb{E}_{x\sim \mu_{\text{ref}}, y\sim \mu_D(x)}[\ln y]$ means written out $E(x):= \int \ln(y)\mu_D(x)$, so in english a function with argument $x$ where $x \in S$ is an picked element in the target Commented Jan 29 at 17:10
• of random variable $X: \Omega \to S$ with distribution $\mu_{\text{ref}}$, right? Commented Jan 29 at 17:13
• Yes you're right since in GAN here the r.v. $y$ is conditioned on the r.v. $x$ which follows the ref distribution, so essentially the law of total expectation applies here. Commented Jan 29 at 17:21