Confused by the latent variables in normalizing flow theory

Question

I'm looking through the paper on variationl inference in normalizing flow and have difficulties with understanding some ideas.

I know there are latent variables $\mathbf{z}_i$ and observed variables $\mathbf{x}$, the final distribution corresponds to $\mathbf{z}_K$, but I can't figure out what observed variables $\mathbf{x}$ are.

Is this initial distribution, for example, Gaussian one? Then why in the paper do the authors call q($\mathbf{z}_0$) initial distribution (Page 3, after expression 7)? I thought q($\mathbf{z}_0$) is the output of the first hidden layer of the network.

The $\mathbf{x}$ is not the weights or hyperparameters of the transformation layers because there are different variables for them, namely $\phi$ and $\theta$ (See page 6, Algorithm 1). Then what is the relation between $\mathbf{x}$ and $\mathbf{z}_0$?

Answer 1

Here is the thing, I think the authors use the notation in eqn.4 in reverse order i.e, $z_L \rightarrow z_{L-1} ... \rightarrow z_1 \rightarrow x$. In this notation $z_L$ has a normal distribution $N(0,I)$ and x is the data distribution whose likelihood is supposed to be maximized. Section 3.1 provides background on the theory of transforming random variables via invertible functions. Now, $z_0$ in this section is some random variable you begin with and then by applying transformations $f_1, f_2, .... f_K$, you get $z_K$ which is another random variable having density given by $q(z_K)$. x is not the hyperparameters of the model, it is the data that you use to learn the model. I believe the Finally, to make the relationship between $z_0$ and $x$ explicit, one can write $z_{0} = f_{1}^{-1}(f_{2}^{-1}(...(f_{K}^{-1}(z_K))..))$ and then $x = f_{0}^{-1}(z_0)$.