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I'm trying to understand and implement the factor-VAE and I'm using the paper Disentangling by Factorising. My probability background is weak and therefore I have issues understanding basic concepts, such as this one:

$\hat{q}(z):=\prod_{j=1}^{d}q(z_j)$

z is in this case the representation given by the encoder, which in my implementation is 10, that is, z$\in \mathcal{R}^{10,1}$. However, it makes no sense, as far as I can tell, to multiply the encoded representations together to get the $\hat{q}$ in my case, since this would correspond to only a scalar and not the vector that I'm interested in. I'd like to believe that each $z_{j}$ comes from a distribution, which may or may not be correlated to the others, and if I sample from all those fictional distributions, I'd somehow end up with $\hat{q}(z)$. Can someone provide me more information regarding this?

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