# Implementation of factorVAE, understanding distributions

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?