A probability distribution of a random variable $\mathbf x$ may be expressed as
$ P(x)=\int P(x|z) P(z) dz$
where $\mathbf z$ is a latent random variable.
In models such as Variational Autoencoders (VAEs) it is common to let $\mathbf z \sim \mathcal{N}(0,I)$ and $P(x|z) = \mathcal{N}\left(x;\mu(z),\Sigma(z)\right)$. However, it is not obvious to me whether those kind of models are able to represent any arbitrary probability distribution $P(x)$ by varying $\mu(z)$ and $\Sigma(z)$, and if that is the case, why it is possible.
Do you have any intuition or references where this is proven?
Thank you very much