# Expected value notation in GAN loss

I am reading Goodfellow's original paper on GANs.

What I struggle to understand is his notation of the subscript in expected values.

$$\mathbb{E}_{\boldsymbol{x} \sim p_{data}(\boldsymbol{x})}\ldots$$

If I understand it correctly then $$\boldsymbol{x}$$ is a realization of some random variable $$\mathbf{x}$$, but how can there be an expectation wrt. $$\boldsymbol{x}$$?

Or is $$\boldsymbol{x}$$ a random variable?

Thank you.

Edit: I do not think this is duplicate, as the referenced question does not answer what $$\boldsymbol{x}$$ means.

• I think it is a duplicate question, but the answer to the duplicate (at stats.stackexchange.com/questions/297158) is not a useful answer! Therefore I have voted to reopen this question, believing it focuses on understanding what the random variable and/or its distribution are, as opposed to understanding what an expectation is.
– whuber
Apr 6, 2019 at 17:40

$$E_{x\sim p(x)}[f(X)]$$ means the expected value of $$f(X)$$ if its assumed to be distributed wrt $$p(x)$$, e.g. for a continuous distribution we have: $$E_{x\sim p(x)}[f(X)]=\int f(x)p(x)dx$$

It's used when the distribution of $$x$$ subject to change in an optimization problem. Specifically, in the paper, authors have two distributions (in page 5) $$p_g$$ and $$p_{data}$$.

Edit: And, the $$x$$ in the subscript of the expected value notation is not a realization. It's the random variable; or more specifically, in the paper it is the random vector, $$\mathbf{x}$$ (It's also in bold in Page 5).

• Thank you. However, I need to specifically know what $x$ is. Is it a random variable or its realization (a sample from distribution)? Apr 6, 2019 at 10:27
• It's definitely not a realization. Realizations have no meaning in that notation., since they don't have such varying distributions (i.e. just constants). And, in the paper it is written in bold and it is the random vector $\mathbf{x}$. Apr 6, 2019 at 10:31
• I strongly suspect the original notation does not refer to a continuous distribution. The subscripted expression "$x\sim p_{\text{data}}$" most likely refers to the random variable determined by sampling from the empirical distribution of the data (with replacement). That is never a continuous distribution. This interpretation also explains the potential for confusion: $\mathbf x$ starts out as the data -- just a collection of numbers -- but it is then used to define a random variable.
– whuber
Apr 6, 2019 at 17:38
• I've chosen the continuous case example arbitrarily to explain the notation. But, in the original paper, $p_g(x)$ and $p_{data}(x)$ indeed refer to continuous distributions (section 4). Apr 6, 2019 at 18:24
• What does assumed to be distributed wrt 𝑝(𝑥). Im finding it hard to wrap my head around that. Is p(x) standard deviation or variance here ? Apr 11, 2019 at 0:38