Need help with notation for objective function

Question

I am reading the paper about Generative Adversarial Nets and there is the equation: $$ \min_{G}\max_{D}V(D,G) = \mathbb{E}_{\mathbf{x} \sim p_{data}(\bf{\mathbf{x}})}[\log D(\mathbf{x})] + \mathbb{E}_{\mathbf{z} \sim p_{z}(\mathbf{z})}[\log(1-D(G(\mathbf{z})))] $$ I am confused by the expectation terms (especially the meaning of the superscript). What does $$ \mathbb{E}_{\mathbf{x} \sim p_{data}(\bf{\mathbf{x}})}[\log D(\mathbf{x})] $$ mean if expressed in plain English? Thanks.

Answer 1

It means the following. If $p_{data}(x)$ is the probability distribution of your data $x$ then ( here S is the region of integration, which I just called S ):

$$ \mathbb{E}_{x \sim p_{data}(x)}[\log{D(x)}] = \int\limits_{S}^{} log[D(x)]p_{data}(x) \mathrm{d}x$$

Usually such notation is used to make it clear with respect to which probability distribution is the expectation computed. Normally you will not see it be used when it is clear from context, but in this case it is not clear from context since on the righ side you are taking the expectation with respect to distribution $p_{data}(x)$ and also with respect to distribution $p_z(z)$