__ Summary and Contributions__: This paper presents completeness results for the general transportability of soft interventions in SCMs. Specifically, the authors continue the line of work on general transportability, extending the work done on atomic/do interventions to support more realistic soft/stochastic/policy interventions and showing that the latter can reduce to the former. The authors present an algorithm to determine necessary and sufficient conditions under which the target effect is computable from a mix of observation data from multiple domains. Finally, the authors graphically characterize soft transportability presenting conditions under which one can infer the lack of transportability visually.

__ Strengths__: This paper does an excellent job of motivating the setting and contributions. The notations are well-defined and compact in one place, making it easy to refer back to when needed. The two illustrative examples used in the paper are helpful in understanding concepts.

__ Weaknesses__: As somebody who has studied causality, has worked with the SCM framework and do-calculus, and to some extent has familiarity with soft interventions, I find myself overwhelmed at times. Please see my comment below on clarity.
Of course, an experiment with simulated data based on the proposed graphical models would make this paper complete.

__ Correctness__: The motivation and situation in historical context is correct. The claims and methods seem not to have any glaring issues. The depth of technical methodology in the main body seems correct.

__ Clarity__: Overall, yes. The paper is well-written and polished.
My only concern regarding the clarity was that the notation (both formulaic and graphical) can be overwhelming at times, even for somebody famililar with the literature. Also, the sudden mentioning of "standard transportability algorithms" in line 181 or "s-Thicket" in line 290 may exude confidence that the authors are familiar with the literature, however, as a reader, this is quite distracting.
On the other hand, I understand that the presented work's contribution would not easily fit within the page limits, and thus I am content with the density of the presented material.
My suggestion would be to rethink whether some parts are necessary in the main body, or move some of the comments that you don't take the time to fully introduce to footnotes.
- fig 2 & 3: the order in which these are presented is confusing and I found myself having to re-read these parts multiple times. Perhaps, lines 127, 128 should go after the description of domains \pi^1, \pi^2 on lines 142-145 and you can merge both figures into (just an idea) a 2-line figure with the first line: Fig 2a, Fig 3a, Fig 3b, and second line: Fig 2b, Fig 2c. This way it's clear what the domains are, it's clear what the intended policy in \pi^* is, and consequently \G^\Delta is immediately inferrable. merging the figures might also save some space, allowing to bring back some of prepositions that were sporadically dropped :)

__ Relation to Prior Work__: To a good extent, yes.
- line 44: missing citations on soft-interventions: Eberhardt. ``Causation and intervention'', and Korb+Hope+Nicholson+Axnick. ``Varieties of causal intervention.''
- line 100: this is something I did not know was considered before: your definition of soft interpretability allows for the post-manipulated str. eqn. to depend on NEW parents (both exogenous or endogenous). Prior definitions that I have encountered only say that the influence of pre-manipulation parents may continue to effect the intervened upon node. Perhaps a brief clarification (even in the footnote) may prevent confusion. relatedly, it seems that the authors are using ";" before soft interventions \sigma, in a sense mirroring the notation of hard interventions as the reserved do-operator. it would be good to clarify this also for the majority who are not familiar with soft interventions.

__ Reproducibility__: No

__ Additional Feedback__:
Scattered nits and questions:
- line 61: why is the distribution of Z in \pi^2 different from that in \pi^*?
- line 65: P^*(Y;...) should be P^*(y;...) for consistency with (1) and later formulations
- line 72: P^*(y;\sigma_X) should be P^*(y;\sigma_X^*) for consistency
- line 108: the subscript of the union writes `X \in \bold{X}`, but it should be `x \in \bold{X}`
- line 136: S_v seems to be undefined (I can't seem to find it in Definition 2 or on page 2)
- line 151: later --> latter
- line 177: is D = d? otherwise D is undefined.
- line 268: there are two Z^1s here; the latter should be Z^2
- line 297: while "heterogenous" is clear after reading the paper in full, introducing this term in the conclusion should likely be avoided
More:
- equation 3: I'm not sure I follow this; where did the W variable go? I understand that the intervention on X removed its direct dependance on W, but can we ignore the bidirectional relation between W and {R,Z}?
- fig 2b: I can't seem to find the description of why the intervention \sigma_X introduces the dependence on R. perhaps add a brief statement about this policy? if it helps, you mention this much later in lines 286, 287 but again without context.
- fig 4: this should really be corrected and made to look consistent with the earlier figures (in the final version of your paper)
------------------------------------------------------------------------------------
Post-rebuttal comment:
I have read the author rebuttal as well as the comments from other reviewers. My review is not changed.

__ Summary and Contributions__: In this work the authors present a formal treatment of transportability settings in the context of soft interventions. They develop necessary and sufficient graphical criteria for deciding soft transportability. Furthermore they develop an algorithm to determine if a non-atomic intervention is computable from a combination of distributions available across domains.

__ Strengths__: This work presents a theoretically grounded solution to the problem of transportability of soft interventions. The solution can in the future perhaps guide the development of practical algorithms.

__ Weaknesses__: While the paper starts with a grand introduction and motivation, it quickly reduces to a bunch of results in the form of lemmas and theorems that are not explained or motivated well. What I find most troubling is that all the proofs (every single one of them) have been moved to the appendix. I wouldn't accept lack of space as an excuse for this because the authors include results like lemma 1 (from a paper in the year 2002) in the main paper that are not essential and could easily be moved to the appendix, and can be well explained with an example.

__ Correctness__: This is a purely theoretical results based paper. However, not even a proof sketch has been provided in the main paper. A good practice for such papers is to provide proofs of sufficiency in the main paper and defer the proofs of necessity to the appendix.
I have not checked all the proofs in the 28 pages long supplementary materials.

__ Clarity__: The first three pages including the introduction are well-written and they explain & motivate the problem well.
Line 84: "as the union of C.." This sentence probably needs to be reworded.
Lines 119-124: It would have been clearer if the letters used to denote variables were in some way related to the variable. For instance use C for credit history instead of W. Also I find the second story pertaining to figure 2 confusing. On the other hand the story pertaining to figure 1 was very well put and easily understandable.
The quality of writing deteriorates page 4 onwards.
Line 139: "while the percentage ought is above"

__ Relation to Prior Work__: I find it hard to believe that [29] is the only related work done by J Robins. It is very likely that the field of epidemiology/public health have more work on this. In fact, figure 1 is a typical problem in these fields.

__ Reproducibility__: Yes

__ Additional Feedback__: 1. Please comment on the part of the review related to prior work.
2. Definition 1 [domain discrepancy]. Note that the term appears only once in the entire paper (i.e. in the definition itself). Also the wording in the definition is sort of confusing.
3. I guess a good understanding of [21] is necessary to understand the surprising element the authors are trying to bring to our attention on page 5 (lines 180-187). Perhaps that part could be written better.
4. Please run spell check.
5. In general, I think there are way too many results crammed into this paper, resulting in the authors having not enough space to clearly explain them.
Updating the review
The paper lacks clarity and is hard to comprehend without a strong understanding of transportability results. My concern above regarding too many results remain true. Also the authors could have done a better job with regard to citing relevant work.

__ Summary and Contributions__: The paper describes a complete algorithm for identifying transportability for soft interventions, as well as a graphical characterization.
I acknowledge reading the author's rebuttal.

__ Strengths__: The paper seems theoretically sound and follows up an important line of work, transportability, that allows one to potentially infer the effects of interventions from existing observational and experimental data, generalizing it to the most general type of interventions, soft interventions.

__ Weaknesses__: While the topic is significant for the general ML community, the paper is very technical and written for a very specific audience, so it maybe hard to read for the general NeurIPS attendee. Also, there is no evaluation section, so it might be a bit more difficult to evaluate the practical impact of the work.

__ Correctness__: As far as I know the paper is correct.

__ Clarity__: The paper is clearly written if one is familiar with the transportability literature, but it might be difficult to follow otherwise.

__ Relation to Prior Work__: The paper is well-positioned and its relationship with the related work is clear.

__ Reproducibility__: Yes

__ Additional Feedback__: A few typos: L39: focuses, L157: particular