# GAN Loss Function Notation Clarification

In the Generative Adversarial Network loss function, what do these mean?: $$E_{x~p_{data}(x)}$$ and $$E_{z~p_{z}(z)}$$ and how are they used in this context?

Generally, the notation $$\mathbb{E}_{x\sim p}f(x)$$ or $$\mathbb{E}_{x\sim p(x)}f(x)$$ refers to the expectation of $$f(x)$$ with respect to the distribution $$p$$ for variable $$x$$ (e.g. see the explanation in the notation section in "Pattern Recognition and Machine Learning" by Bishop).
In the context of GANs, it means that for an objective function such as $$\mathbb{E}_{x\sim p_{data}(x)}[logD(x)] + \mathbb{E}_{z\sim p_z(z)}[log(1 - D(G(z)))]$$ the first summand is the expectation with regards to $$x$$ coming from the data and the second summand is the expectation with regards to $$z$$ which you sample from $$p_z$$ as an input for G.
Since the first summand stands for correctly classified real data (coming from $$p_{data}$$) and the second summand stands for correctly as fake classified images coming from G using $$z$$ sampled from $$p_z$$, D tries to maximize this expression. And G does the opposite.