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In the VAE, we approximate the probability distribution $p(z \mid x)$, where $z$ is the latent vector and $x$ is our data. The reason is that $p(z \mid x)$ becomes impossible to calculate for continuous data because of $p(x)$, which require integration (not in closed form) to be solved.

But why don't we also need to approximate $p(x \mid z)$?

What I can guess here is that, in VAEs, we assume $p(z)$ (prior), so we are able to calculate $p(x \mid z)$, but for $p(x)$ we can't assume its distribution? Is it right?

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What I can guess here is that, in VAEs, we assume $p(z)$ (prior), so we are able to calculate $p(x \mid z)$, but for $p(x)$ we can't assume its distribution? Is it right?

You could assume $p(x)$ is some distribution (e.g. the Gaussian $\mathcal{N}(0, 1)$), but that assumption might be completely wrong - you can also check that against your given data (i.e. if your data is unlikely under your assumed distribution, then maybe your assumption is unreasonable - see this or this).

The ultimate goal of a generative model, like the VAE, is to be able to generate data (so you want to use it for tasks like image denoising). So, you want to learn a probability distribution that approximates the distribution from which your given dataset was sampled that would allow you to sample more similar data. In practice, you want to learn a probability distribution of the form $p(x, z)$, where $z$ is some latent random variable, which we assume to be involved in the process of generating $x$ (a random variable that represents the type of data that you have), i.e. $x$ depends on $z$.

By definition, $p(x, z)$ can be written as $p(x, z) = p(x \mid z)p(z)$ or $p(x, z) = p(z \mid x) p(x)$. If you put together these equations, you obtain the Bayes rule/theorem $p(z \mid x) = \frac{p(x \mid z)p(z)}{p(x)}$, where $p(z \mid x)$ is the posterior, $p(x \mid z)$ the likelihood, $p(z)$ the prior, and $p(x)$ the marginal (likelihood) or evidence.

In the VAE, we use the variational distribution, $q_\phi$, to approximate $p_\theta(z \mid x)$, but we also learn $p_\theta(x \mid z)$ - in fact, that's the decoder - and we learn it jointly with the variational distribution by maximizing the ELBO. We maximize the ELBO because it's a lower bound on $p(x)$, so, by maximizing it, we're also maximizing the marginal likelihood (aka evidence) $p(x)$, i.e. we're trying to find the parameters $\theta$ and $\phi$, such that $x$ is more likely to have been sample from $p(x)$, which is unknown. See the VAE paper (in particular, algorithm 1) for more details.

Once the VAE is trained, you have $q_\tilde{\phi}(z \mid x)$ and $p_\tilde{\theta}(x \mid z)$. If $p_\theta(z \mid x)$ is the posterior, then $q_\tilde{\phi}(z \mid x)$ is an approximation of the posterior. Now, you can generate data with the VAE as follows

  1. Let $q_\tilde{\phi}(z \mid x)$ be the new prior (as opposed to the user-defined $p(z)$, which was used to train the VAE) - in Bayes statistics, the posterior then becomes the prior (it's our new belief of how $z$ behaves given our data $x$)
  2. Sample $z^i$ from $q_\tilde{\phi}(z \mid x)$
  3. Sample $x^i$ from $p_\tilde{\theta}(x \mid z^i)$

So, together, $\color{blue}{p_\tilde{\theta}(x \mid z^i)}$ and $\color{red}{q_\tilde{\phi}(z \mid x)}$ approximate $p(x, z) = \color{blue}{p(x \mid z)} \color{red}{p(z)}$.

It might be possible to come up with a new objective function (different from the ELBO), where we use some kind of variational distribution to approximate $p_\theta(x \mid z)$, but, in the VAE, that's not the case.

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