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I have question about a formula in the machine learning paper. The paper is as follows.

https://arxiv.org/pdf/1906.02691.pdf

In page 9, formula (1.6), I totally agree with it since it is famous formula in the Prof.Koller's book "probabilistic graphical model". The formula is as follows.

$p_\theta(x_1,...,x_M) = \prod\limits_{j=1}^Mp_\theta(x_j|Pa(x_j))$      (1.6)

However, conditioned on x, I cannot agree with the formula (2.2) in page 16 which is as follows.

$q_\phi(z|x) = q_\phi(z_1,...,z_M|x) = \prod\limits_{j=1}^Mq_\phi(z_j|Pa(z_j),x)$      (2.2)

I think $q_\phi(z_j | Pa(z_j),x)$ should be changed to $q_\phi(z_j | x)$ since $Pa(z_j)$ equals to $x$. This refers to the case where the graphical model is $x$ --> $z$.

Even if the graphical model is not $x$ --> $z$ but $x$ --> $v$ --> $z$ which implies $Pa(z_j) = v$ by introducing temporary variable $v$, I think formula (2.2) still has problem. In this case, I think the original formula (2.2) should be changed to follows.

$q_\phi(z|Pa(z_j),x) = p_\phi(z_1,...,z_M|Pa(z_j),x) = \prod\limits_{j=1}^Mq_\phi(z_j|Pa(z_j),x)$

If someone can derive the formula (2.2), please help me. Thank you.

vorton
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  • In my opinion you have to see x in Eq. 2.2 not as a r.v. but as a parameter. So you have a graphical model with just Z-random variables involved, parametrically dependent on x. – Thomas May 25 '22 at 20:34
  • Anyway in the VAE the distribution q(z|x) is assumed to be an indpendent product of unidimensional Gaussian, so that at the end the usual VAE comes from a disconnected (trivial) graphical model. (the mean and variance of the Gaussians depend on x), so I am not sure why this generic formulation – Thomas May 25 '22 at 20:36
  • Maybe you can also compare with the original article of Kigma and Welling https://arxiv.org/abs/1312.6114 – Thomas May 25 '22 at 20:37
  • ( saying thx for the input you receive is good manner by the way ) – Thomas May 29 '22 at 12:31
  • I was busy on sth. I'm still confusing why the formula is like that. Anyway, Thank you so much!! – vorton Jun 10 '22 at 06:32

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