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As far as I am reading about VAE, I always see a graphical model of Z --> X,

I know P(X) can be intractable, but that would be if we would have many dependencies. But here is only Z. So why variational inference at first place?:)

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I'm not sure if I get the question correctly, but we use variational inference, because we do have many dependencies. X and Z are usually high-dimensional vectors with complex relationships.

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Exact inference is possible only when your latent variables can take only a discrete set of values. But the computation required grows quite fast with an increase in the dimensionality of latent space. As you change your latent space to a continonus space, the inference becomes intractable.

Hence Inference is the key algorithmic problem. And our goal is to build General and scalable approaches to inference.

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