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a normal autoencoder decodes single latents z to single images (or whatever other kind of data) x, and also encodes single images x to single latents z.
with VAEs, we want our decoder (p(x|z)) to take single latents z and output a distribution over x's. for simplicity we generally declare that this distribution is a gaussian with identity covariance, and we have our decoder output a single x value that is the mean of the gaussian.
because each x can be produced by multiple z's, to run this backwards you also need a distribution of z's for each sin... (read more)