# Why does the GAN paper make the assumption that $G(Z) = X$?

There exists crucial assumption in one (several) proof of the GAN paper that doesn't really make sense to me.

Let $$X \sim p_{data}$$ be the random variable associated with the input data, and $$Z \sim p_{Z}$$ be the random variable associated random noise. Note that they are written with bold letter in the paper e.g. $$\mathbf{x}$$.

In the computation of the optimal discriminator, in Proposition 1, the paper found that, https://papers.nips.cc/paper/5423-generative-adversarial-nets.pdf

$$D^{*}(X) = \frac{p_{data}(X)}{p_{data}(X) + p_{g}(X)}$$

where $$p_{g}(X)$$ is the probability distribution of the output of the random noise to the generator, $$G(Z).$$

However, to make this calculation work, the paper makes the key assumption that $$X = G(Z) \sim p_g$$.

That is, the random variable associated with the data $$X$$ is the same random variable as $$G(Z)$$.

However, in my opinion this is not true.

First, while $$G(Z)$$ represents a random variable takes on value in the same space as the data, they are is not exactly the same space. This means, $$X: \Omega_1 \to \mathcal{X} \subseteq \mathbb{R}^n$$, but $$G(Z): \Omega_2 \to \mathcal{Y} \subseteq \mathbb{R}^n$$, where $$\mathcal{Y}$$ could be a super or subset of $$\mathcal{X}$$. Furthermore, the sample spaces $$\Omega_1, \Omega_2$$ associated with the two random variables may differ as well. All this is to say that the random variables $$X$$, $$G(Z)$$ are not the same, hence we cannot make the argument that $$G(Z) = X$$, and proceed to calculate the optimal discriminator as shown in Proposition 1.

Also, at the notation level this is also troublesome, because $$X \sim p_{data}$$ is the random variable representing the data, but now $$X \sim p_g$$ as well.

All the above problems can be solved by denoting $$G(Z)$$ using a different random variable, say $$X^\prime = G(Z)$$. But the authors did not make this decision.

Therefore, I don't understand how equation (3) is derived.

Can anyone help me with this question?

I think the problem here is one of notation. The notation $$X \sim p_G$$ as used in the GAN paper is not meant to indicate $$X$$ is distributed according to $$p_G$$, but rather sampled from $$p_G$$. Of course this raises the problem about why the following equality holds in the proof:
$$\mathbb{E}_{Z \sim p_Z}\left[ \log (1-D(G(z))\right] = \mathbb{E}_{X \sim p_G}\left[ \log (1-D(X)\right]$$
This is effectively a change of variables provided $$G^{-1}$$ exists. If not, one can use the result of the Radon–Nikodym theorem to change the measure.