Answers.Body: <p>It's not supposed to be derived from some equation. That is the basic premise under which GANs work. <em>The output of the Generator <span class="math-container">$G(z)$</span> is fed as an input <span class="math-container">$x_g$</span> to the discriminator</em>.</p>
  
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2019-06-25T19:28:17.233Z
Answers.LastActivityDate: 2019-06-25T19:28:17.233Z
Answers.UserDisplayName: Djib2011
Body: <p>In the proofs for the original GAN paper, it is written:</p>
  
  <p><span class="math-container">$$∫_x p_{data}(x) \log D(x)dx+∫_zp(z)\log(1−D(G(z)))dz
  =∫_xp_{data}(x)\log D(x)+p_G(x) \log(1−D(x))dx$$</span></p>
  
  <p>I've seen some explanations asserting that the following equality is the key to understanding:</p>
  
  <p><span class="math-container">$$E_{z∼p_z(z)}log(1−D(G(z)))=E_{x∼p_G(x)}log(1−D(x))$$</span></p>
  
  <p>which is a consequence of the LOTUS theorem and <span class="math-container">$x_g = g(z)$</span>. Why is <span class="math-container">$x_g = g(z)$</span>?</p>
  
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2019-06-17T20:14:09.867Z
Comments.Text: in all formulas shown you never showed $x_g$ or $g(z)$. What do these refer to and where?
Comments.UserDisplayName: mshlis
Tags: <neural-networks><machine-learning><math><generative-adversarial-networks><proofs>
Title: How is G(z) related to x in GAN proof?
UserDisplayName: NNLearner
ThreadText: # How is G(z) related to x in GAN proof?
  
  (Asked by: username_0 on 2019-06-15)
  
  In the proofs for the original GAN paper, it is written:
  
  $$∫\_x p\_{data}(x) \log D(x)dx+∫\_zp(z)\log(1−D(G(z)))dz
  =∫\_xp\_{data}(x)\log D(x)+p\_G(x) \log(1−D(x))dx$$
  
  I've seen some explanations asserting that the following equality is the key to understanding:
  
  $$E\_{z∼p\_z(z)}log(1−D(G(z)))=E\_{x∼p\_G(x)}log(1−D(x))$$
  
  which is a consequence of the LOTUS theorem and $x\_g = g(z)$. Why is $x\_g = g(z)$?
  
  **Tags:** neural-networks, machine-learning, math, generative-adversarial-networks, proofs
  > **Comments:**
  > **username_1**: in all formulas shown you never showed $x_g$ or $g(z)$. What do these refer to and where?  
  
  ---
  
  ## 1 answers
  
  ### Answer by username_2 on 2019-06-25. Score: 1 | Accepted answer
  
  It's not supposed to be derived from some equation. That is the basic premise under which GANs work. *The output of the Generator $G(z)$ is fed as an input $x\_g$ to the discriminator*.
  
  
  ---
  
  