Understanding Event-Generation Networks via Uncertainties

1. Detailed explanations for the features seen
2. Thorough investigation of the uncertainties in BINNs

1. Lots of small grammatical errors, in terms of punctuation and proper use of plural

The authors study the uncertainties associated with invertible neural networks (INN) through the use of promoting the deterministic weights to distributions. These weights are then updated through the use of Bayesian statistics. This study opens a new approach to study the accuracy of the INNs, and to help pinpoint regions of phase space that need more data to accurately reproduce the probability distribution.

Overall, the article is for the most part very well written, with a few places in which the authors are unclear detracting from the reader's understanding of the work. Additionally, there are a fair number grammatical errors, but these do not subtract from the overall presentation of the material.

The authors take the time to study the features that are seen in their toy models in such a way to help gain some insight for real world results. The authors provide sufficient details of the hyper-parameters to allow the reader to reproduce their work.

The requested changes can be found in the section below. With the changes below, I believe that this article is suitable for publication in SciPost Physics.

1. Page 3, in the paragraph starting: "One contribution to the error budget are systematic ...", the authors should change this to read "Two contributions to the ...," and the authors should switch the subscripts for $\sigma_{th/sys}$ to $\sigma_{sys/th}$ to be consistent with the previous part of the sentence.
2. In the same paragraph as 1, the sentence starting, "For example accounting for large, momentum-dependent logarithms ..." is unclear in what the authors are trying to convey. I would recommend reworking this sentence.
3. On page 4 and page 15, the authors use the word "effectively," but the word "efficiently" would be better suited in these situations.
4. Page 8, the authors state that the "always remove the phase space boundaries," but never specify what they consider the boundary. Is it just the exact endpoint, or some region around each endpoint, or something else?
5. Page 9, in the sentence "If we now assume..." the authors have the sentence fragment "as it will turn out useful," in which the meaning is unclear. Do they mean that the fit will be useful or the interpretation of the network acting like a fit? The authors should clarify this sentence fragment.
6. Page 10, the two sentences after equation 23 can be combined into a single sentence to make the meaning more clear. The first sentence is clear enough, but the second sentence is not clear.
7. Page 12 and page 13, the authors reference the lower panels of Fig. 4 and Fig. 6 respectively, but the Figures are laid out horizontally and not vertically. The text should be changed to reflect this.
8. Page 12, in the same sentence as point 7, the authors mention a "2-parameter fig," I believe this should read "2-parameter fit."
9. Page 13, in the same sentence as point 7, the authors show a one-parameter fit, but spend the previous few sentences discussing the two different contributions and when different ones dominate. They should either do a two-parameter fit, or give some justification on why they do not include the second parameter.
10. Page 17, the authors mention in two separate places a comparison between Fig. 8 and Fig. 4. It would be beneficial for the reader for the authors to overlay the curves from Fig. 4 on Fig. 8 to make their points more clear and easy for the reader to see.
11. Overall, the paper could benefit from a quick read through to check for appropriate use of plural verbs with plural nouns, and other minor grammatical errors that do not take away from the overall message.