Question Regarding the Normalizing Flow Equation

[JasonWUCHI]

Dear author,

I am trying to understand the equation 4 in the paper, could you help me to clarify some point?

1. Should it be $\bar{\theta}=f\phi(z)$ instead of $\bar{\theta}=f\phi^{-1}(z)$
2. How do you deal with the $f_{\phi}^{-1}$ term when you calculate from line 2 to line 3?

Thank you in advance!

[saidwivedi]

Thank you interest in our paper :)

1. $\mathcal{f}_{\phi}$ is the normalizing flow network which transforms a simple distribution $z$ to . Since the network is invertible $\mathcal{f}_{\phi}^{-1}$ and we are interested to model the error $\bar{\theta}$, hence $\bar{\theta} = \mathcal{f}_{\phi}^{-1}(z)$
2. I not sure if I understood the question fully. We replaced $\mathcal{f}_{\phi}^{-1}(\bar{\theta_{g}})$ with $(\theta_{g} - \theta)/\sigma$ and omitted for simplicity.

This formulation is adapted from RLE, ICCV 2021 which defined residual log-likelihood. You can refer to the original paper if it helps.