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From my understanding:

Variational autoencoders sample the latent variables $y$ using a proposal distribution $q$ of the observed variables $x$. The objective is that the decoder $p$ applied to $y$ defines a distribution whose value at $x$ is supposed to be maximized (reconstruction cost) plus a regularizer term.

Why not forget about $q$, and for every example $x$, iteratively update $y$ so as to maximize the objective? That is, setting $y$ to

$$\arg \max_y P(x | y)$$

rather than sampling it from $q(y \mid x)$.

Is $q$ an approximation motivated by inference speed?

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Typically, we use the prior $p(y) = \mathcal{N}(0, I)$ and choose $p(x|y) = \mathcal{N}(\mu, \sigma = f(y;\theta))$ for some neural network $f(\cdot; \theta)$. The ELBO (evidence lower bound) is the objective of the VAE:

$$\log p(x) \geq E_{y \sim q(y|x)}[\log p(x|y)] - \mathcal{D}_{\text{KL}}(q(y|x)||p(y))$$

iteratively update $y$ so as to maximize the objective?

I'm not completely sure what you mean by "update $y$". There's no single $y$ value in our model to update, since it is a latent variable. We can only update our variational approximation of the posterior distribution of $y$ -- which we do by learning the parameters of the model of $q$.

Is q an approximation motivated by inference speed?

Yes, while training, we would like to sample $y$ from the true posterior distribution $p(y|x) = \frac{1}{Z} p(y)p(x|y)$ -- this would eliminate the gap between the ELBO and the true evidence $\log p(x)$ -- the inequality would become an equality. Unfortunately, sampling from this distribution is computationally intractable, so we resort to an approximation.

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  • $\begingroup$ I've updated my question to explain what I mean by iteratively update $y$. For many objectives, this $y$ is pretty easy to find. $\endgroup$ – Neil G Feb 17 at 7:12
  • $\begingroup$ @NeilG ah I see what you mean. But in that case, you would be optimizing something which is no longer the ELBO, and thus no longer a lower bound on $\log p(x)$, which is our goal $\endgroup$ – shimao Feb 17 at 7:20
  • $\begingroup$ That's what I'm asking. Ultimately, we want the best value of $y$, which is not the same as the one that optimizes ELBO. Why don't we just go ahead and find that. That's our real goal after all. $\endgroup$ – Neil G Feb 17 at 7:42
  • $\begingroup$ @NeilG no, the ultimate goal of generative modeling is to learn the distribution of our data $x$ -- one way to operationalize this is to minimize the KL-divergence between our model $p(x)$ and the data distribution $D$, which is equivalent to maximizing $E_{x \sim D}[\log p(x)]$, which the ELBO gives us a lower bound on. we don't "actually" care about finding the value of $y$ which produces the best reconstruction -- reconstruction loss just happens to show up in the ELBO. $\endgroup$ – shimao Feb 17 at 9:09
  • $\begingroup$ I agree with you up to the point where you say we're minimizing $E_{x\sim D} \log p(x \mid y)$. However, this $y$ can be defined using the distribution $q(y \mid x)$, or alternatively using the procedure I suggested. What I am asking for is why not use my procedure instead of q. What your answer is doing is restating the procedure that q uses. I understand that procedure. What I don't understand is why it's better than just using my procedure. $\endgroup$ – Neil G Feb 17 at 9:44

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