We appreciate your valuable comments and suggestions. We address each of them below.

1. ** $Experimental comparison$ ** Yes, we completely agree that our work could be strengthened by experimentally comparing with other clustering methods, especially if other methods require any special assumptions that are not present in ours (like the margin-type conditions as in k-means) to be employed in the causal clustering framework. However, this would require a closer analysis at those methods. If other alternatives do not require particularly stronger assumptions and can be readily integrated into the causal clustering framework as well (though this must be theoretically verified as in our work), then the problem turns to which approach to use for specific data. Because each clustering algorithm has benefits and drawbacks, it is difficult to conduct a fair comparison using experiments. Given these circumstances, we believe that designing a fair experiment to demonstrate the superiority of our proposed methods is not quite straightforward, and that's outside the scope of this paper.

   Furthermore, comparisons with other "causal clustering" counterparts in the SCM literature appear unclear because they are designed to analyze structural heterogeneity, whereas ours is designed to analyze treatment effect heterogeneity. We may consider some special data generation process where we may demonstrate superiority of our method, yet this might not be convincing.

   All in all, we believe that confining our work to presenting the respective theory that the two appealing off-the-shelf cluster-analysis methods can be successfully adopted within the novel framework of causal clustering would suffice for the time being. Nonetheless, following your comment, we will go over the potential extension to other clustering algorithms in the discussion section in the revised manuscript.

2. ** $Ans to Q1$ ** This is a good question. The rate of convergence of regression functions in nonparametric modeling is well studied (e.g., Györfi, 2002). It depends on both the function space to which the true $\mu_a$ belongs and the estimator itself $\hat{\mu}_a$; e.g., for $\mu_a$ in the Holder class with smoothness $s$, given $dim(X)=d$, the best (in minimax sense) rates are $O(n^{-\frac{s}{2s+d}})$. If $\mu_a$ is not smooth enough, then high-dimensional covariates will increase uncertainty of our estimators.

   In fact, in our simulation in Section 5.1., we incorporated the effect of the number of covariates by directly controlling the convergence rate through a parameter $\beta$, i.e., by letting $\Vert \hat{\mu}_a - \mu_a\Vert = O_P(n^{-\beta})$. The results are presented in Figure 2 (we will magnify this figure in the revised manuscript).

3. ** $Ans to Q2$ ** In our opinion, there is no precise answer to this question. As previously stated, the smaller the number of covariates ($=d$), the lower the estimation error. However, additional components also interact with $d$. For example, in density-based clustering, we have bandwidth that is entangled with $d$, both of which affect the error. More importantly, if we use only a small number of covariates, the no-unmeasured confounding assumption (Assumption C2) is likely to be violated; we usually collect as many covariates as possible to ensure there is no nmeasured confounders left. All of these factors must be considered simultaneously.

4. ** $Ans to Q3$ ** This leads to an entirely different, but interesting question. Given a collection of covariates, one can accurately estimate the CATE function using approaches such as Kennedy (2023), and then apply an algorithm to determine which covariate (or combination of covariates) has the most impact on the treatment effect. Or based on our methods, one can analyze the covariate distributions across clusters and determine which covariate results in the greatest distributional divergence in subgroup effects. We will add this comment in our revised manuscript.

5. ** $Interpretation$ ** This is also a great question. One possible interpretation is that each cluster has its own subpopulation from which units are generated. Properties of each subpopulation could be studied by analyzing distributional features, etc. However, as we highlighted in the Discussion section as a shortcoming, while our approach allows for efficient discovery of subgroup structures,, it may be less effective for prescriptive applications (i.e., informing specific treatment decisions). Thus, it is important to exercise caution while attempting to interpret the observed clusters.

We appreciate your valuable input and insights. We address each of your comments below.

1. ** $Clarification on contributions$ ** Thank you for bringing our attention to this. The main contributions of our work is that we have proved that the two appealing off-the-shelf cluster-analysis techniques can be successfully adopted within the novel framework of causal clustering, using simple plug-in methods without requiring additional strong structural assumptions as opposed to k-means, which requires the margin condition (kim et al. 2024). For example, in Section 3, we demonstrate that the robust inductive hierarchical clustering method (Balcan el al. 2014) may be applied to causal clustering through the plug-in estimator with minimal assumptions, and we analytically validate the associated costs in Theorem 3.1. Verifying this is not as straightforward as one may imagine. We will state our contributions more clearly in Section 1, particularly in relation to kim et al. (2024) and key references in Sections 3 & 4.

2. ** $Experimental comparison$ ** Relatedly, yes, we completely agree that our work could be strengthened by experimentally comparing with other methods such as hdbscan, if other methods require any special assumptions that are not present in ours (like the margin-type conditions as in k-means) to be employed in the causal clustering framework. However, this would require a closer analysis at those methods. If other alternatives do not require particularly stronger assumptions and can be readily integrated into the causal clustering framework as well (though this must be verified), then the problem turns to which approach to use for specific data. Since each clustering algorithm has its own pros and demerits, it is difficult to conduct a fair comparison using experiments. Given these circumstances, we believe that designing a fair experiment to demonstrate the superiority of our proposed methods is not quite straightforward, and that's outside the scope of this paper. (Please let us know if we misinterpret your intention here). Following your comment, however, we will provide a brief discussion of the potential extension to other clustering algorithms.

3. ** $Plug-in KDE$ ** As pointed out, our plug-in estimator essentially inherits pros and cons of the standard KDE (in level-set clustering). In theory and practice, the choice of the kernel affects the performance of KDE through the bandwidth, and when the bandwidth is appropriately chosen, the shape of the kernel has little effect on the performance. In kernel density estimation (KDE), Gaussian kernel has the advantage that the number of modes monotonically decreases as the bandwidth increases (Silverman, 1981, Using kernel density estimates to investigate multimodality). In kernel regression, Epanechnikov kernel is the most efficient in the constant term of mean integrated squared error, but other usual kernels (including Gaussian) are at least ~90% efficient. Other than these effects, the shape of the kernel does not affect the convergence rate and has little effect on learning error by less than ~10% on the constant. The choice of kernel has no effect on our results (but it may in real data studies).

4. ** $Issues with presentation$ ** Thank you for pointing these out.

   • We shall remove the $\equiv$ signs, which had been often used to emphasize notational "equivalence," and replace them with standard equal signs.

   • We totally agree it should be better to broke down the definition 3.1 as in the original work of Balcan et al. Will do that accordingly in the revised manuscript (or at least in the appendix).

   • Thank you for correcting the typos; during revision, we will fix everything and proofread thoroughly.

5. ** $Ans to Q1$ ** As you have mentioned, the Hausdorff distance $H(S_{1}, S_{2})$ is oblivious to the density but only sensitive to the envelope: given that we already have enough points in a specific region, adding more points does not meaningfully change the Hausdorff distance. However, we are looking at the level sets $L_{t,h}=\\{w\in \mathbb{R}^{q}: p_{h}(w)>t\\}$ for $t>0$, and their estimators $\hat{L}\_{t,h}$ for $t>0$. Then the density information is encoded through the data points in the level sets. For example, if one distribution has high density regions but the other does not, then for a large $t>0$, the level set ${L}\_{t,h}$ for the first distribution would be nonempty, while the level set for the second distribution would be empty. Hence, although the Hausdorff distance is oblivious to the density, the density is encoded through the level sets $L\_{t,h}$, and measuring the difference by the Hausdorff distance behaves sensitively with respect to the density of data points.

6. ** $Ans to Q5$ ** We believe your insight is correct. We don't yet have a good answer to how to reconcile with the related structural causal learning tasks. As we leave comments for reviewer STix, causal clustering using the SCM approach may yield different results from ours. We will at least comment on this in the discussion section, with the hope of opening up new options for subsequent research.

7. ** $Ans to Q6$ ** You are correct - just like Balcan et al (2014) our estimator requires tuning the two noise parameters. We do not propose any good solution for parameter selection in our paper. Nonetheless, Balcan et al. have empirically showed robustness to such parameter tuning, thus we believe our method can inherit this property, albeit further work is needed because these parameters play an essential part in our method. We will add this discussion in the revised work.

Thank you very much for your valuable feedback. As you suggested, we will add a new paragraph on Q5 and Q6 as the limitation of our work in the discussion section, hoping it opens a new avenue for future research. We are confident that incorporating your suggestions above will significantly enhance the quality of our paper.

Thank you for your time and thorough feedback. We have addressed each of your concerns as outlined below.

1. [Originality and related work] Thank you for highlighting the connection between our research and other works in the literature on causal discovery or structural causal models. To our knowledge, there are three common ways to express causal/counterfactual quantities: (1) structural equations, (2) causal graphs or structural models, and (3) potential outcomes (counterfactuals). While these languages can complement each other, the target parameters, assumptions, notations, and techniques often differ. We acknowledge that the prior works you mentioned also address the notion of "causal clustering." However, they fall into the second approach (in the context of learning structural heterogeneity), whereas ours is based on the third (learning treatment effect heterogeneity). This distinction is why we did not include the extensive body of work in causal discovery/structural causal models. To the best of our knowledge, Kim et al. (2024) were the first to formally study the task of analyzing treatment effect heterogeneity via clustering based on the (potentially multivariate) CATE within the potential outcomes framework. In our revised manuscript, we will clearly highlight this difference and define our "causal clustering" problem. Additionally, based on your comments, we will add a separate subsection in Section 1 explaining the connection between our approach and others in the causal structure learning literature.

2. [Figures and Presentation] Thank you for pointing this out. We agree that some figures, particularly Figure 4, are too small. Since the essential idea to be presented is quite macroscopic (i.e., cluster patterns), we believe that expanding it slightly larger with increased font and legend size can alleviate this issue. We will address this in the amended text, and if necessary, we will move some figures to the Appendix.

3. [Inappropriate verbatim from [36]] We appreciate your suggestion regarding this matter. Our reliance on the previous work of Kim et al. (2024) is mainly for describing the motivation, problem, and setup in Sections 1 and 2. The framework is novel within the community, and our intention is to present the problem accurately without distorting its description. (As mentioned above, our problem differs significantly from the causal clustering framework in the causal structure learning literature.) However, we acknowledge that we should avoid using verbatim as much as possible and rephrase where necessary. We will fully address this in the revised text.

   Nonetheless, we would like to emphasize that Sections 3 and 4, which constitute the main contribution of our paper, are entirely original and do not contain any issues related to verbatim content from [36]. Therefore, in our opinion, this should not significantly diminish the value of our paper.

4. [Ans to Q1] We apologize: it should be a distance with the image $[0,1]$. We will fix this.

5. [Ans to Q2] We apologize for the confusion. As you noted, it is an erroneous expression, and part of the sentence should be revised as follows: "... with respect to the true target clustering, because we build a set of nested clusters across various resolutions (a hierarchy) such that the target clustering is close to some pruning of that hierarchy."

6. [Ans to Q3] These could be utilized, for example, to develop precision medicine or optimal policy. We will add specific examples in the discussion section.

7. [Ethics Review] This is a very good point. In the last section, we will discuss how the discovered subgroup was formed simply based on similarity in treatment effect, without considering factors such as fairness/bias. We believe that discovering a "fair subgroup" would be an intriguing future project of our work.

If there are any remaining issues, please feel free to let us know through your comments.

We appreciate your helpful questions and comments, and we feel that the revised manuscript will be significantly improved as a result. Most importantly, we appreciate the reviewer pointing out the issue of several inappropriate verbatims while we define the problem and setup in Section 2. We take this issue extremely seriously and will fully address it in the revised manuscript. Please feel free to let us know if you have any further concerns or questions.

Thank you for your time and valuable feedback. We would like to address each of your major concerns in the following.

1. [Novelty of the problem] Thank you for pointing this out. First, we want to emphasize that the problem of causal clustering is novel; clustering on counterfactual outcomes provides a new framework for analyzing treatment effect heterogeneity, and as far as we know this approach has not addressed (or at the very least, formally addressed) in the literature before, except in Kim et al. (2024). It also differs significantly from other previous attempts in the cluster analysis literature that considered partially observed outcomes or clustering with measurement errors, since in our case the variable to be clustered consists of completely unknown counterfactual functionals (See Section 1.2 of Kim et al. (2024)). Due of the page limits, we just briefly describe the problem in Section 2 and refer readers to Kim et al. (2024) for details, which could confuse them about the problem's novelty. Based on the reviewer's comment, we will explicitly clarify on the problem, setting, and associated challenges.

2. [Contribution] We would like to stress out that our main contribution is not on the algorithm side, i.e., the use of the robust hierarchical or density-based clustering algorithms. Rather, it is on the theoretical side where we provide conditions under which the plug-in estimator works for each algorithm for causal clustering framework.

   • The problem of causal clustering poses challenges which do not appear in the previous studies, as we cluster on unknown counterfactual functionals that is to be estimated nonparametrically. Surprisingly, a plug-in approach does NOT always works without strong extra conditions. Kim et al. (2024) showed that, in the case of k-means, even the plug-in estimator will fail without the margin condition, which requires local control of the probability around the Voronoi boundaries; without such structural assumptions, the error cannot be bounded.
   • The main contributions of our work are found in Sections 3 and 4, where we prove that the two appealing off-the-shelf cluster-analysis techniques can be successfully adopted in the framework of causal clustering using simple plug-in methods without requiring extra strong structural assumptions as opposed to k-means, which requires the margin condition (kim et al. 2024). As outlined in the proof, verifying this is more complicated than one may think. (We will give a brief exposition why this could be a theoretically difficult task in the revised text.) Our plug-in approaches could also be readily extended to clustering with generic pseudo-outcomes, even outside the context of causal inference. This versatility may be considered an additional contribution.

3. [Presentation] As mentioned earlier, and as suggested by the reviewer, we will ensure that our revised paper is more self-contained. We believe this will help clarify the novelty of the problem and our key contribution, ensuring that readers do not experience any confusion.

4. [Ans to Question] Thank you for this insightful question. There are two types of error we consider in our work: estimation error of the unknown counterfactual functionals (i.e., the identified regression functions $\{\mu_a\}$) and error regarding to clustering accuracy. I guess your question is related to the latter, or more specifically, whether one may adopt other types of clustering algorithms that can incorporate uncertainty on the hierarchy that is being found. We completely agree that this could lead to fascinating future work, and we will include it in the revised manuscript's discussion section.

We hope the above response addresses your concerns. However, if there are any remaining issues, please feel free to let us know through your comments.

The reviewer has read the response and thanks tha authors for their time. However given the presentation issues, and the weaknesses raised, the reviewer still thinks the paper should be improved before publications and that it is not ready as is.

Thank you for your input. We believe incorporating your comments on presentation will improve the quality of the paper.