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On page 11 of this VAE tutorial it is said that new samples of the data distribution X can be found by plugging z ~ N(0, I) into the Decoder P.

I don't understand why this is true. During training, in order for f(z) to approach X in distribution, z was sampled from N($\mu(X), \Sigma(X)$) [as seen on page 10]. So how does sampling from ~N(0, I) at test time work.

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It makes sense because the first term in the loss (in the ELBO), is the KL divergence between N(0,I) and the N($\mu(X), \Sigma(X)). During training, you essentially have a trade off between reconstructing perfectly and constraining the distribution to have a certain prior (here standard normal) in order to be able to generate new sample using this prior.

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