General response to all reviewers and the AC

We are grateful to all reviewers for their time and the useful feedback and suggestions. We are pleased that reviewers consider our work to tackle ”an important problem” (hDXV), “well-written” (hDXV, nez5), “mathematically rigorous” (qoW1), and found the empirical results “strong” (nez5) and “effective” (aTiY). Two reviewers also highlighted the proposal of new, provably consistent estimators as a key strength (hDXV, aTiY).

A concern of reviewers hDXV and qoW1 is the applicability of the proposed methods. For the former, we refer to Appx. B, where we provide relaxations for some of the conditions over the number of categories of $W$ and $U$. For the latter, we clarify in our response that our model can accommodate arbitrary shifts in the distribution of $U$, even without changes in $N_U$, via the domain indicator $E$. We thank reviewer nez5 for pointers to related work on data fusion and transportability, which we will incorporate in Section 1.1. We will also include an evaluation of MAE for the WAdj and WAdj* baselines (for which no CIs are available), as well as a runtime analysis for the proposed methods to the baselines, following suggestions by reviewers nez5 and aTiY, respectively.

Our responses to individual reviewers provide further details. We believe to have thus addressed all reviewers’ concerns, but remain available for further discussion and clarification.

Individual response to reviewer

Thank you for your time in reviewing our work and for your questions and remarks which have helped us to clarify some key aspects of our setting.

Your identification result states that the causal effect can be recovered from (W, X, Y) in the source domain and W in the target domain. Are there any typos? I believe it should be (E, W, X, Y).

Thank you for pointing out this possibly confusing formulation. What we mean here is that ``the causal effect can be recovered from $\{P(W,X,Y|E=e_1), …, P(W,X,Y|E=e_{k_E})\}$ and $Q(W)=P(W|do(E:=e_T))$’’, i.e., we do indeed require the domain labels. (The main point we wanted to convey here is that observations of $X$ and $Y$ are not needed in the target domain.) We will adjust the formulation accordingly to clarify.

Your identification result appears to contradict my intuition. The causal effect depends on U, W, and N_Y, where both U and W follow different distributions in the source and target domains. It is therefore not intuitive that the causal effect could be recovered using only W.

It may perhaps be unintuitive at first, but this is precisely the main insight of proximal causal inference: a causal effect can be identified, even in the presence of unobserved confounding $U$, if sufficiently informative proxies $W$ are observed. This insight is not specific to our work, but also central to the works described in Section 1.1, many of which are by now well-established in the causal inference literature. While it is true that the joint and marginal distributions of $U$ and $W$ may change across domains, a crucial aspect of our setting is that the way in which $W$ depends on $U$---captured by the conditional distribution $P(W|U)$ (or $P(W|U,Z)$ for the more general setting discussed in Appx. B)---remains invariant across domains. This invariance (which follows from the modularity of causal models) is leveraged to transform the adjustment formula from Eq. (3) such that it only involves observable quantities, see also l.130-132. Imagine that $W$ is a perfect copy of $U$, i.e., $W=U$ and $P(W|U)$ is the identity. In this case, observing $W$ in the target domain suffices, because the terms in Eq. (3) become $P(y|W,x)$ and $Q(W)$. Our identification result shows that this argument extends to noisy or imperfect copies of $U$, provided that they are sufficiently informative as formalised by Asm. 1. In particular, Asm. 1 rules out cases such as that of a completely uninformative proxy $W$ that is independent of $U$, for which identification would become impossible. Intuitively, due to the invariance of other conditionals, observations of an informative proxy $W$ in the target domain can suffice to correct for changes in the distribution of $U$.

The source of my concern lies in your specification of E and U. Your setting assumes that all factors responsible for shifts in the distribution of U across domains are fully observed through E. This is a very strong assumption, and if it is indeed being made, it should be stated explicitly and clearly in the text. If N_U differs between the source and target domains, then your identification result does not hold. Is that correct? If that is the case, then the applicability of your result in practical settings may be substantially limited.

We believe there may be a misunderstanding regarding our use of structural equations for modelling how and which distributions change across domains. We will attempt to clarify this key aspect of our setup.

SCMs and induced distributions. The noises $N_i$ in Eq. (1) are exogenous (not being modelled) and only serve to introduce stochasticity into the otherwise deterministic model. The variables of interest are $E,U,W,X,Y$. Crucially, the functions $f_i$ and noise distributions $P_{N_i}$ are completely unconstrained. Hence, the SCM can induce any $P(E)$, $P(U|E)$, $P(W|U)$, $P(X|U)$, and $P(Y|U,W,X)$, see, e.g., Sec. 3.4 of Peters et al. [42] for details. In particular, the distributions of $U$ can change in arbitrary ways across domains, i.e., $P(U|E=e_1)$, …, $P(U|E=e_{k_E})$, $Q(U)$ are (a priori) unconstrained. While we use SCMs to formalize the data-generating process, the paper could also be written purely in terms of conditional pmfs (see l. 100-101).

The role of the environment variable $E$. The variable $E$ is not assumed to explicitly capture “factors responsible for shifts in the distribution of U”. In line with prior literature on causal transportability and domain adaptation, $E$ is simply a domain indicator (e.g., $E \in \{1, 2, 3, 4\}$ for experiments run in different labs) that can be thought to switch between different distributions for $U$. We only assume that these distributions are stable within any given domain (i.e., for a fixed value of $E$), which is a standard assumption for many well-established methods. As explained above, the distributions of $U$ can, in general, differ arbitrarily across domains, even without $N_U$ changing; they are only implicitly constrained through Asm. 1.

Identifiability with Shifting Confounder Distributions. To answer your direct question: ``If $N_U$ differs between the source and target domains, then your identification result does not hold. Is that correct?'' No, this is incorrect. Our identification result is agnostic to why the distribution of $U$ changes. The key requirement is that the rank condition, $\text{rank}(P(W|E,x)) \geq k_U$, is met in the source domains. The mechanism that generates $U$ can change arbitrarily. As explained above, the standard assumption that $N_U$ does not change across domains comes without loss of generality, but changes to $N_U$ could equally be accommodated without causing issues. We will revise the manuscript to clarify these crucial points and make the role of $E$ as a simple indicator variable and the relation between Eq. (1) and changes to $P(U)$ more explicit. In light of these explanations, we hope that you will agree that our modelling of the relationship between $E$ and $U$ does not limit the applicability of our results.

In relation to the concerns mentioned above, I also have reservations regarding the applicability of your results in practical settings. In your application, E represents the hotel ID. Can all factors responsible for shifts in the distribution of U truly be captured by E alone?

We hope that our previous response also clarifies this issue. [To briefly reiterate the main point: Even without changes to the distribution of $N_U$, arbitrary changes in $P(U)$ can be modelled via $U:=f_U(E,N_U)$ via a domain indicator $E$. One way to see this is inverse cumulative distribution function (CDF) sampling. Consider two domains with arbitrary distributions $P_1(U)$ and $P_2(U)$. Let $N_U$ be uniform and define $f_U(e_1, n_U):=F_1^{-1}(n_U)$ and $f_U(e_2, n_U):=F_2^{-1}(n_U)$ where $F_1$ and $F_2$ are the CDFs of $P_1$ and $P_2$. This construction induces $P_1$ in $e_1$ and $P_2$ in $e_2$. The key is the flexibility of the function $f_U$, so that different values of $E$ can index different functions of the second (stochastic) argument.]

Is it necessary to specify e_T for estimation? In the application, E represents hotel IDs, and the results change when the target hotel ID e_T is replaced. This behavior is quite strange.

$E$ is simply a categorical identifier for distinct domains. In the application, each hotel ID is an arbitrary label. The model does not use the value of the ID itself in any calculation. The key point is that changing the target ID from, say, Hotel A to Hotel B, means we are asking a different question: "What is the causal effect for Hotel A?" versus "What is the causal effect for Hotel B?". These environments have different characteristics—specifically, different distributions of the proxy variable $W$ (price). That is, changing the target hotel ID also changes $Q(W)$. Our method leverages this information from the proxy in the target domain to transport the causal effect correctly. Therefore, the estimate should change. If the results did not change when switching the target from a luxury hotel to a budget hotel, it would imply the model is failing to account for the target-specific context. The observed behavior thus confirms that the model is working as intended.

We hope that our responses have clarified some misunderstandings and resolved your concerns. In light of this, we kindly ask that you reconsider your evaluation.

Thank you for the continued feedback and for reconsidering your evaluation. We address your remaining questions below.

Based on your settings, which mechanisms are variable and which are invariant to the environment? Which conditional causal effects are invariant to the environment? As with the distributional literature, lists of variant and invariant sets of probabilities are helpful. For example, Covariate shift: The distribution of input variables P(X) changes, but the conditional distribution P(Y|X) remains the same.

The mechanisms that may vary are those for which $E$ appears as an argument on the RHS of the corresponding structural equation in Eq. (1); the others remain invariant. (This follows from the Markov property, see l.99-100.) In terms of the induced conditional probability distributions, this means that:

• $P(U)$, or rather $P(U|E)$, varies across domains
• $P(X|U)$, $P(W|U)$, and $P(Y|W,U,X)$ remain invariant

While this may seem like a rather strong assumption at first, it is important to consider that $U$ is unobserved. When restricting our attention to observable variables (as is common practice), there are no invariances, i.e., the entire joint distribution $P(X,Y,W)$ can vary across domains. In particular, $P(X)$, $P(Y|X)$, $P(Y)$, and $P(X|Y)$ may all change. In this sense, the latent shift assumption is weaker than, e.g., covariate shift, label shift, or conditional shift (which all require some form of invariance involving only observed variables).

Regarding invariant conditional causal effects: Since only $P(U)$ changes across domains, any causal effect that conditions on $U$ will be domain-invariant. However, because $U$ is unobserved, this does not have direct practical implications. If we again consider conditional causal effects involving only observed variables, generally none of them are domain-invariant. This precisely necessitates our approach for correcting or transporting conditional causal effects to the target domain.

We will try to make these points clearer in the updated descriptions of prior work (Sec. 1.1) and our setting (Sec. 2).

How can you justify the setting in your application?

As mentioned in your initial review, many factors could conceivably confound the causal effect of interest, such as the location or service quality of the hotel. Our setting allows for shifts in the distribution of these factors (as summarised in the form of a categorical variable $U$) but requires that the other conditional distributions given $U$ remain invariant. For example, invariance of $W|U$ implies that different hotels that share the same characteristics $U=u$ (location, service quality, etc) will also exhibit the same price distribution.

Our assumptions from Fig. 1(a) likely do not perfectly match the setting at hand. As shown in Appx. C.2, though, our results also hold for more general graphs with additional covariates or edges, which may be more appropriate. Our objective was to evaluate the developed estimators for the simple setting presented in the main paper. Our empirical results from Fig. 4 and the MAEs reported in our response to reviewer nez5confirm the effectiveness of our estimator compared to baselines, despite likely misspecification and violations of some of our assumptions.

General response to all reviewers and the AC

We are grateful to all reviewers for their time and the useful feedback and suggestions. We are pleased that reviewers consider our work to tackle ”an important problem” (hDXV), “well-written” (hDXV, nez5), “mathematically rigorous” (qoW1), and found the empirical results “strong” (nez5) and “effective” (aTiY). Two reviewers also highlighted the proposal of new, provably consistent estimators as a key strength (hDXV, aTiY).

A concern of reviewers hDXV and qoW1 is the applicability of the proposed methods. For the former, we refer to Appx. B, where we provide relaxations for some of the conditions over the number of categories of $W$ and $U$. For the latter, we clarify in our response that our model can accommodate arbitrary shifts in the distribution of $U$, even without changes in $N_U$, via the domain indicator $E$. We thank reviewer nez5 for pointers to related work on data fusion and transportability, which we will incorporate in Section 1.1. We will also include an evaluation of MAE for the WAdj and WAdj* baselines (for which no CIs are available), as well as a runtime analysis for the proposed methods to the baselines, following suggestions by reviewers nez5 and aTiY, respectively.

Our responses to individual reviewers provide further details. We believe to have thus addressed all reviewers’ concerns, but remain available for further discussion and clarification.

Individual response to reviewer

Thank you for your time reviewing our work and your suggestions to extend our section about related literature. We address each point individually below.

It is not clear to me why WAdj and WAdj* are missing from the real data experiment.

The reason for excluding WAdj and WAdj* from the real data experiment was the lack of a method for calculating confidence intervals for these estimators, which would reduce the relevance of the comparison presented in Figures 4 and 10. Wald or exact binomial test confidence intervals cannot be used for WAdj and WAdj*, as opposed to the Oracle and NoAdj* estimators, and there is not a simple derivation as in the case of the reduced estimator. The main alternative would be to consider bootstrap confidence intervals. To compare the point estimates of WAdj and Wadj* with the rest of the methods in the real data experiment, we have computed the mean absolute error (MAE) with respect to the oracle during this rebuttal period obtaining the following values: 0.044 (Reduced), 0.051 (NoAdj), 0.080 (NoAdj*), 0.053 (WAdj) and 0.075 (WAdj*). This suggests that the proposed reduced estimator performs better in terms of absolute error than all the considered baselines in this real data example. We will include these results in Section 6 of the main paper.

I feel like it is overlooking important work on e.g. data fusion in causal inference.

Thank you very much for the pointers. We agree that prior works on data fusion and transportability, combining observational and experimental studies, and causal approaches to multi-domain data are related and will include additional paragraphs in Sec. 1.1. To briefly comment on some commonalities and differences: The work of Pearl, Bareinboim et al. on data fusion considers a similar problem. However, in that line of work, transportability is typically determined purely from the extended causal graph (a.k.a. selection diagram) and (extensions of) do-calculus. In contrast, proximal causal inference is less fully nonparametric and typically relies on additional constraints such as Asm. 1 (e.g., requiring invertibility of certain matrices, thus formalising sufficiently informative proxies). In our setting, identifiability thus cannot be concluded solely from the causal graph, but leverages additional assumptions of a different nature than those explored in the transportability literature. Work on combining experimental and observational data typically aims to overcome the statistical challenge of small sample size in the experimental study, e.g., by devising biased but lower variance estimators that incorporate observational data. In contrast, in our setting, we do not have any observations of $X$ or $Y$ in the target domain (taking the role of the experimental data), but only observational data from different domains. Causal methods for multi-domain data often seek to learn an invariant predictor by discarding spurious features. Like the most closely related work by Tsai et al., they typically target the conditional mean $E_Q[Y|x]$, rather than the interventional distribution $q(y|do(x))$.

In the real data experiment, for some cases the reduced estimators disagrees with the oracle and the NoAdj is actually closer to the oracle, for instance target domain id 3 or 11. Do you have an hypothesis for why this happens in this particular cases? I wonder if any of the assumptions you make are particularly violated here.

The NoAdj and NoAdj* estimators do not consider the existence of an unmeasured confounder $U$. Therefore, in those domains in which the causal effect of $X$ on $Y$ is barely confounded, these estimators can obtain good estimates in terms of absolute error. Even though the change of domain does not directly affect the equations for $X$ and $Y$ in the SCM, the strength of the confounding can change due to the distribution shifts of $U$ when changing the domain. Furthermore, the NoAdj* estimator, which is the one that performs really well in terms of absolute error for the target domains ids 3 and 11 uses an i.i.d. sample $(X_1,Y_1),\dots,(X_n,Y_n)$ from the target domain, and this information is not available in our setting. Therefore, NoAdj* has access to some data that our proposed estimators do not (see Appendix D.3 for additional details).

General response to all reviewers and the AC

We are grateful to all reviewers for their time and the useful feedback and suggestions. We are pleased that reviewers consider our work to tackle ”an important problem” (hDXV), “well-written” (hDXV, nez5), “mathematically rigorous” (qoW1), and found the empirical results “strong” (nez5) and “effective” (aTiY). Two reviewers also highlighted the proposal of new, provably consistent estimators as a key strength (hDXV, aTiY).

A concern of reviewers hDXV and qoW1 is the applicability of the proposed methods. For the former, we refer to Appx. B, where we provide relaxations for some of the conditions over the number of categories of $W$ and $U$. For the latter, we clarify in our response that our model can accommodate arbitrary shifts in the distribution of $U$, even without changes in $N_U$, via the domain indicator $E$. We thank reviewer nez5 for pointers to related work on data fusion and transportability, which we will incorporate in Section 1.1. We will also include an evaluation of MAE for the WAdj and WAdj* baselines (for which no CIs are available), as well as a runtime analysis for the proposed methods to the baselines, following suggestions by reviewers nez5 and aTiY, respectively.

Our responses to individual reviewers provide further details. We believe to have thus addressed all reviewers’ concerns, but remain available for further discussion and clarification.

Individual response to reviewer

Thank you very much for your time. We answer your individual questions below.

The paper does not seem to provide a method on finding any valid W if one exists (in polynomial time), or a method on enumerating all valid W.

Thanks for the remark. We will address what we consider as two separate aspects: choosing among known proxies and finding new ones.

Regarding how to select the best proxy from a known set of valid proxies, our framework does not require a selection. Instead, these proxies can be combined into a single, more informative proxy (i.e., consider the combined proxy that takes values in the Cartesian product of the sets of values the individual proxies can take). If any of the proxies satisfy Asm. 2, then the combined proxy will satisfy it as well. Therefore, the problem of selection is sidestepped by combination.

Regarding how to efficiently find or enumerate all valid proxies $W$ in a known large, complex causal graph, we agree with the reviewer that this is an interesting and distinct research problem. The focus of our paper is on establishing identifiability and estimation given a valid proxy. Developing a polynomial-time search algorithm to check the graphical conditions listed in Remark 8 in Appx. B is a valuable direction for future work but falls outside the scope of our current contribution. If the causal graph is completely unknown, it does not appear to be testable whether a valid proxy exists. Instead, domain expertise is needed to argue for the validity of a given proxy (similar to justifying instrument validity in IV models).

It could be useful to have a general runtime analysis, in a form of O() analysis (if applicable), or across different controlled variables (e.g., n) and compare the proposed estimators against other baseline estimators, rather than on one specific instance as shown in Footnote 5, page 7.

Thank you for the suggestion. Since our main focus was not on computational cost of the proposed methods, we only included a specific instance (Footnote 5) to show that it is not really computationally expensive. The main factor that increases the runtime is the dimension (number of categories) of the categorical random variables. However, we will look into this further and will try to augment the manuscript with a O() analysis or more comprehensive runtime numbers.

Is the identification formula shown in Eq. (4) of Theorem 1 complete? In other words, is it the case that W is the proxy variable (as defined in Assumption 1) if and only if q(y|do(x)) is identified with Eq. (4)?

Yes, we believe the identification formula in Equation (4) to be complete. Thm. 1 shows that $q(y|do(x))$ is identified under Assumption 1. Conversely, if $q(y|do(x))$ is identified with Eq. (4) for any form of the confounder, then the right pseudoinverse of $P(W|E,x)$ exists and, therefore, this matrix has linearly independent rows. Furthermore, in this case, it can be shown that the inequality $k_W \geq k_U$ is obtained as follows from Equation (4):

$\forall Q(U), P(y|U,x): \quad P(y|U,x) Q(U) = P(y|E,x) P(W|E,x)^{\dagger} P(W|U) Q(U) \Rightarrow$

$\forall P(y|U,x): \quad P(y|U,x) = P(y|E,x) P(W|E,x)^{\dagger} P(W|U) \Rightarrow$

$\forall P(y|U,x): \quad P(y|U,x) = P(y|U,x) P(U|E,x) P(W|E,x)^{\dagger} P(W|U) \Rightarrow$

$I_{k_U}= P(U|E,x) P(W|E,x)^{\dagger} P(W|U) \Rightarrow$

$k_U = rank(I_{k_U}) \leq \min( rank(P(U|E,x)), rank(P(W|E,x)), rank(P(W|U)) ) \leq k_W$

General response to all reviewers and the AC

We are grateful to all reviewers for their time and the useful feedback and suggestions. We are pleased that reviewers consider our work to tackle ”an important problem” (hDXV), “well-written” (hDXV, nez5), “mathematically rigorous” (qoW1), and found the empirical results “strong” (nez5) and “effective” (aTiY). Two reviewers also highlighted the proposal of new, provably consistent estimators as a key strength (hDXV, aTiY).

A concern of reviewers hDXV and qoW1 is the applicability of the proposed methods. For the former, we refer to Appx. B, where we provide relaxations for some of the conditions over the number of categories of $W$ and $U$. For the latter, we clarify in our response that our model can accommodate arbitrary shifts in the distribution of $U$, even without changes in $N_U$, via the domain indicator $E$. We thank reviewer nez5 for pointers to related work on data fusion and transportability, which we will incorporate in Section 1.1. We will also include an evaluation of MAE for the WAdj and WAdj* baselines (for which no CIs are available), as well as a runtime analysis for the proposed methods to the baselines, following suggestions by reviewers nez5 and aTiY, respectively.

Our responses to individual reviewers provide further details. We believe to have thus addressed all reviewers’ concerns, but remain available for further discussion and clarification.

Individual response to reviewer

Thank you for your time in reviewing our work. We respond to your questions and comments below.

To enable identifiability of P(y|do(x)) in the target domain, the authors require that the number of domains k_E is bigger than the number of realizations k_U, k_W of variables W and U, respectively and additionally k_W should be equal to k_U. This assumption is unrealistic, and almost never happens in practice. Therefore, I am not sure how significant the result is.

Thank you for this question. It addresses an important part of our work. We agree that Assumption 1, which implies $k_W = k_U$, is restrictive. However, this is a simplified version of our assumption, presented in the main text for clarity. We provide a more general and weaker assumption in Appx. B (we mention this in Sec. 3.1). To avoid confusion, we decided that for the revised version of our manuscript, we will only state the general result of Appx. B directly in the main part of the paper.

Clarification of the Necessary Assumption. Our results only require the rank condition $rank(P(W|E,x)) \geq k_U$ to hold (see Appx. B). This condition is less restrictive than the one highlighted by the reviewer and only implies that $k_E, k_W \geq k_U$. Importantly, this rank condition is also a necessary condition for identifiability. If $rank(P(W|E,x)) < k_U$, it is, in general, not possible to identify the causal effect. To illustrate, consider the case where $E$ and $W$ are only correlated with the first $k_U - 1$ values the confounder $U$ can take. In this scenario, the last value, $u_{k_U}$, can act as a hidden confounder with no proxy, making $P(Y | \text{do}(X))$ non-identifiable. More precisely, assume we have an SCM with hidden variable $\tilde{U}$ such that $rank(P(W|E,x)) = k_U-1$. Now modify this hidden confounder such that $U \coloneqq \tilde{U} 1(N=1) + u_{k_U} 1(N \neq 1)$, where $N$ is independent of $E$ and $W$, to get a new SCM. If in the new SCM $X$ and $Y$ are correlated through $U = u_{k_U}$, then this confounding cannot be detected by $E$ or $W$ as the joint distribution of $(E, W)$ does not change between the two SCMs. More generally, if the rank condition is violated, there will always be a linear combination of the confounder's values whose influence on $Y$ and $X$ cannot be detected through $E$ or $W$. We will formalize this intuitive argument in the revised manuscript.

Significance. Achieving identifiability in the presence of unobserved confounders is a fundamentally challenging problem that always requires strong assumptions. We contend that our more general assumptions in Appendix B, particularly the generalized rank condition, is no stronger than those made in comparable works that achieve identifiability. In the revised version of our manuscript, we will expand our discussion to better contextualize our assumptions within the broader literature.

How does the estimator that is presented in Section 4.1 relate to Theorem 1?

Thm. 1 is a theoretical result establishing that the causal effect is identifiable under our assumptions. While the theorem's statement directly suggests a "plug-in" estimator (i.e., by estimating the matrices on the right-hand side), such an approach is not necessarily the only possible estimator. We therefore also propose an estimator based on Maximum Likelihood of the full causal structure, building off Prop. 2. The connection to Thm. 1 is that our proof of consistency for this estimator (Prop. 3) requires the identifiability result from Thm. 1. In short, the theorem guarantees a unique target exists, which our proposed estimator is proven to converge to.

Is the identifiability of P(y|do(x)) in Section 4.1 also requires the assumption 1?

As discussed in our answer to the previous question, the identifiability result presented in Sec. 3 is required for the consistency result of the estimators in Sec. 4.1. For the revised manuscript, we will make this connection clearer.

The assumption that only the equation of unobserved variables changes across the environments is also seemed to be too strong and rarely observed in practice.

We thank the reviewer for this comment, as it highlights a point we need to make clearer in the paper. Our method is not limited to the setting where only the unobserved variables change. This was a simplifying assumption used for the illustrative example in the main text, chosen for pedagogical clarity and its connection to previous work [30]. In fact, our theoretical guarantees hold for a much broader class of SCMs. In our manuscript we provide a detailed discussion and examples of these more general cases---where, for example, the mechanisms for observed variables like $X$ also shift across environments---in Appendix C.2 (Figure 5). The key requirement for our method is not the immutability of specific equations, but rather the structural properties that allow for identification.

I thank the authors for their responses; most of my concerns were resolved, and I will increase my score to 4. Also, I will give a few comments that I have after reading the author's response:

• Even if $k_W=k_U$ is just a simplified version of the assumption presented in the main work, still, my main concern was regarding the assumption $k_E \geq k_U$. Via simple words, it requires observing the data in the different domains/environments, the number of which is not less than the size of the domain of the unobserved variable $U$. This solely sounds to me very unrealistic. But it is nice to have a theoretical result that claims that this is a necessary condition for the identifiability.
• Our method is not limited to the setting where only the unobserved variables change. This is very confusing, because the equations (1) at the beginning of Section 2 define the data generating process in which only the generating process of $U$ is domain dependent. If this is only a simplification, I would ask the authors to make it more explicit in the main part or at least make some reference to the Appendix, where this assumption is simplified.