# Confused by the latent variables in normalizing flow theory

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 $$\mathbf{z}_0$$?

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 $$x = f_{0}^{-1}(z_0)$$.
• Thanks, but the paragraph 4.2 says that $q_0$ is Gaussian and it is initial density Jul 1, 2020 at 20:26