3. The heuristic nature of the decoupling approach in the proposed method

Thank you for your critical observation regarding our decoupling approach. We understand your concern about the potential heuristic nature of this approach. However, Our approach in PoNet is grounded in a hypothesized causal graph, from which we establish an inference policy. This policy is then operationalized using a data-driven model to decouple the covariates into three factors: confounders and two types of post-treatment variables. The particular structure of our model is designed as multi-channels based on the treatment assignment, to ensure that the learned representation of confounders is pre-treatment, while the other two factors are post-treatment. Moreover, through the supervised information from the data label and the backpropagation during training, we further ensure that the inferred confounders and mediation post-treatment variables are predictive to outcome while collider post-treatment variables are irrelevant to the outcome. Then the reconstruction loss ensures that the learned representations retain the essential information from the original covariates, while the MIMR helps in encouraging the independence of the representations of confounders, mediation post-treatment, and collider post-treatment variables. This method aims to balance the need for information preservation and the separation of different types of variables, which is a challenging task in high-dimensional and complex datasets. Moreover, our extensive experimentation, detailed in the paper, demonstrates the effectiveness of our proposed model in evaluation. PoNet consistently outperforms state-of-the-art models across various scenarios, indicating its robustness and reliability in handling post-treatment bias. This empirical evidence supports the efficacy of our method, suggesting that, despite potential theoretical concerns, PoNet is a reliable tool for addressing post-treatment bias in practical applications.

1.Practical applicability of assumptions in real-world scenarios

Thank you for your valuable insight regarding the assumption in our study. We recognize that in many real-world scenarios, domain knowledge and prior information play a crucial role in identifying confounders and post-treatment variables. We recognize that in many real-world scenarios, domain knowledge and prior information play a crucial role in identifying confounders. However, our assumption is grounded in the reality that there are numerous complex scenarios where such clear distinctions are not readily apparent. Especially in emerging fields or interdisciplinary research, the intricate interplay between various factors might not be fully understood. For example, in the field of investigating the role of microbiota in Chronic Diseases, particularly gut microbiota, the exact role and mechanisms through which microbiota influence chronic diseases like obesity, diabetes, and heart disease are still under investigation, with many conflicting studies and theories. In the psychological field, there is ongoing debate about which psychotherapy methods are most effective for certain disorders and why they work. Different schools of thought (e.g., cognitive-behavioral, psychodynamic) offer differing explanations and approaches, and the field is constantly evolving with new research. In such cases, the ability to differentiate between confounders and post-treatment variables can be significantly challenging. This is particularly true in large-scale observational studies where the volume and variety of data can obscure the underlying causal structure.

Moreover, our approach is designed to provide a robust framework for scenarios where domain knowledge is limited or where the causal relationships among variables are not well understood. By allowing for the possibility that the nature of covariates is not fully known, our model offers a potential and general solution against potential post-treatment biases that might arise from the observational data. This can be particularly valuable in exploratory analyses or in fields where causal relationships are still being investigated.

In summary, while we acknowledge that domain knowledge is invaluable in identifying confounders in many situations, our approach caters to those complex and less understood scenarios where such clarity is not available. The flexibility of our model to adapt to both well-understood and ambiguous situations enhances its practical applicability across a wide range of real-world problems.

2. Established methods for discovering causal relationships might be more direct

Thank you for your insightful question regarding the use of causal discovery in the context of unknown causal structures in observational data. While it is true that conditional independence tests are a valuable tool in causal discovery, particularly in identifying v-structures induced by post-treatment variables, they also come with certain limitations, especially in complex data environments. For instance, when dealing with high-dimensional covariates, the computational cost of these methods can be quite substantial. Moreover, causal discovery based on conditional independence tests faces the challenge of Markov equivalence classes. This issue means that for some causal structures, it is not possible to determine the direction of causality solely through independence tests. Additionally, many of these methods tend to be linear and may struggle to model causal relationships in high-dimensional and complex environments effectively.

Our proposed method, PoNet, offers an alternative approach by focusing on identifying and distinguishing the distributions of three different types of covariates – confounders, collider post-treatment variables, and mediation post-treatment variables. As we demonstrate in Theorem 1, if we can successfully identify and separate these distributions from the covariates, the causal effect can be identified in the presence of post-treatment variables, without the need to fully understand the specific causal relationships between covariates.

But we deeply appreciate the value of your suggestion regarding the use of causal discovery. Leveraging causal discovery in the estimation of causal effects without predetermined causal structures represents an exciting and promising direction for future exploration in this field.Thank you once again for your insightful suggestion.

Dear Reviewer,

I hope this message finds you well. I am writing to follow up on the response we recently submitted addressing the concerns you raised regarding our manuscript, for which you had assigned a score of 5. This score, while moderately favorable, indicates that there were aspects of our work that could be improved.

In response to your valuable feedback, as well as that of other reviewers, we have undertaken significant revisions to our manuscript. These revisions include extensive additional experiments and new theoretical analyses, which have substantially strengthened the overall quality and contribution of our work. We believe that these enhancements have addressed the key areas of concern and have brought our paper more in line with the high standards of the conference. Your expertise and feedback are highly valued, and we would greatly appreciate your reassessment of our revised submission.

Thank you for your consideration, and we look forward to any further insights or feedback you may have.

Best regards,

Authors

3. Applicability of a variable decomposition CATE estimator in the collider case.

Thank you for your question. As analyzed in our paper, in the scenario of collider post-treatment bias, if an estimator does not separate collider post-treatment variables from the original covariates and excludes these variables from the outcome estimation, collider bias would still persist. This holds true even for decomposition CATE estimators such as TEDVAE. Regarding your observation that TEDVAE performs best among all compared baselines on the real-world dataset, it's important to note that although TEDVAE does not specifically account for post-treatment scenarios, its objective is to decompose three distinct factors: confounders, risk factors (which only affect the outcome), and instrumental factors (which only affect treatment and do not affect the outcome). In the context of collider post-treatment bias, one of our goals is to learn a factor $Z_c$ that does not affect the outcome, which aligns with the instrumental factors in TEDVAE: For the prediction of the outcome, both the collider factor $Z_c$ and the instrumental factor can be considered as noise. Both PoNet and TEDVAE effectively exclude these factors when predicting the outcome. Consequently, this approach may contribute to TEDVAE's superior performance on the real-world dataset compared to other baseline methods. In summary, while TEDVAE and similar estimators do not directly address post-treatment scenarios, their methodological approach to factor decomposition aligns with the objectives of learning non-affecting factors, which can inadvertently contribute to their effective performance in the presence of post-treatment bias.

4. The potential of a generative approach accommodating variable decomposition and post-treatment covariates.

Thank you for your thought-provoking question regarding the potential of a generative approach to address post-treatment bias. We acknowledge the merit in considering a generative approach for handling post-treatment bias. Indeed, generative models, which establish joint distributions between input and output data, can offer comprehensive insights into data relationships, potentially providing a robust framework for addressing complex issues like post-treatment bias. However, it's important to recognize that generative models come with their own set of assumptions and requirements, particularly regarding the distribution of data. They necessitate assumptions and modeling of the data distribution, which can be challenging for complex datasets, especially when computational resources are limited. The strong assumptions required for generative models may not always be suitable for smaller-scale datasets or for those with intricate distribution characteristics. Despite these challenges, we agree that exploring post-treatment bias from a generative perspective is a promising avenue. This approach could potentially offer novel insights and solutions to this pervasive issue in causal inference. We are grateful for your suggestion and intend to further explore this direction in our future work. This exploration would contribute to the ongoing efforts in the field to develop more versatile and effective methods for addressing various biases in causal inference.

1.Limited technical contribution and similarity to existing methods.

Thank you for your comments regarding the technical aspects of our proposed algorithm. We acknowledge that in terms of the model architecture, we have not made extensive alterations from the TARNet framework, because architecture design is not our focus. However, we would like to emphasize that the structure design of the model is not the primary contribution of our work. Our significant contribution lies in unifying two different sources of post-treatment bias within a single framework and distinguishing between confounders and two types of post-treatment variables, which is rarely considered in the field of treatment effect estimation from observational data. In addition, our work importantly advances the theoretical understanding of causal effect identifiability under these complex conditions. We have rigorously proved that, given certain causal assumptions, causal effects are identifiable even in the presence of both confounders and two distinct types of post-treatment variables. Besides, in the current version we also enrich this work's theoretical contribution by proving the inferred confounders and post-treatment factors are minimally sufficient (It refers to a statistic that captures all the information in a sample that is relevant for estimating a parameter of the underlying probability distribution.) for the needed parameter $\theta$ for the true treatment effect, more details can be found in Appendix C.

2. Performance of PoNet on datasets without post-treatment covariates.

Thank you for raising the question about the applicability and performance of PoNet in scenarios where datasets may not have post-treatment covariates. In response to your suggestion, we have conducted additional experiments using a new semi-synthetic dataset generated from PeerRead, where post-treatment bias has been removed. The performance on error of PEHE, ATE and MSE on the test data is shown in the following table, we also have updated it in the Appendix:

The results indicate that PoNet is not only adaptable to this situation but also maintains its efficiency, often achieving state-of-the-art results compared to other methods. Essentially, if the dataset only contains pre-treatment covariates, PoNet will focus on adjusting for confounders to estimate the treatment effect accurately .Furthermore, our model’s inferred​ $Z_c$, which is excluded to predict outcome, in the absence of post-treatment bias, essentially acts to remove noise variables from the original covariates. This feature contributes to PoNet's superior performance in this scenario. The results from these experiments on the PeerRead dataset without post-treatment variables demonstrate that PoNet can effectively handle both post-treatment and confounding biases, showcasing its robustness and versatility.

7. More guidance for practitioners and extending experiments.

Thank you for your insightful suggestion to provide more guidance for practitioners and to extend our experiments to scenarios where post-treatment bias is not a significant issue. In response to your suggestion, we have conducted additional experiments using a new semi-synthetic dataset generated from PeerRead, where post-treatment bias has been removed. The performance on error of PEHE, ATE and MSE on the test data is shown in the following table, we also have updated it in Table 8 in the Appendix:

These experiments have been enlightening, showing that PoNet is adept not only at eliminating post-treatment bias but also at addressing confounding bias,which is achieved through the balancing module in PoNet, which adjusts for the confounders .Furthermore, our model inferred​ $Z_c$, which is excluded to predict the outcome, in the absence of post-treatment bias, essentially acts to remove noise variables from the original covariates. This feature contributes to PoNet's superior performance in this scenario. The results from these experiments on the PeerRead dataset demonstrate that PoNet can effectively handle both post-treatment and confounding biases, showcasing its robustness and versatility.

While we fully acknowledge and support your idea of comparing the performances of different methods on standard benchmark datasets without post-treatment bias, due to time constraints and the extensive workload required, we plan to undertake this additional experimentation in future work. However, we believe the results from our extended experiments on the PeerRead dataset already provide evidence of PoNet's efficacy in various scenarios to a certain extent. We appreciate your suggestion and are committed to further exploring these avenues to enhance the utility and applicability of PoNet in the broader causal inference landscape.

8. Applicability details of PoNet (running times, resource needs).

Thank you for your inquiry regarding the practical applicability details of PoNet, including running times and hardware requirements. For the experiments conducted in our study, we utilized an Intel(R) Xeon(R) Gold 5120 CPU @ 2.20GHz with 512GB of RAM, and an NVIDIA TITAN RTX GPU with 24 GB of memory. The datasets used in our experiments varied in size, with the synthetic dataset having dimensions (15000, 24/48/72), the semi-synthetic dataset (7601, 1020/1153/1344), and the real-world dataset (6133, 69). Focusing on the computational aspects, taking the largest semi-synthetic dataset used in our experiments with size (7601, 1344) as an example, the PoNet model exhibited efficient performance. On average, each epoch of training took approximately 1.1 seconds without batch partitioning. The peak GPU memory consumption was around 1500MB.

As for the requirement of sizable datasets, our experiments indicate that PoNet performs effectively on the datasets of the sizes we tested. We believe PoNet to be adaptable to various dataset sizes, although the optimal performance is subject to the complexity and nature of the data. We hope this information assists practitioners in understanding the applicability and resource requirements of PoNet and facilitates its adoption in relevant fields.

9. about the first sentence of section 3.3 is incomplete

Thank you for pointing this out. The sentence should be: Separating confounders and post-treatment variables is essential for unbiased treatment effect estimation. We have corrected it in the revised version.

5. Consideration of error on ATE in addition to CATE.

We are immensely grateful for your constructive suggestion to consider the error on ATE, in addition to the CATE, as an evaluation metric in our study. Acting on your valuable feedback, we have re-conducted our experiments to include the error metrics of ATE on the synthetic and semi-synthetic datasets in our evaluation. For example, the updated results regarding to the ATE metric on semi-synthetic PeerRead are as follows:

The results of these additional experiments are now incorporated in our paper. Specifically, the results on the synthetic and semi-synthetic datasets are presented in Table 6 and 7 in Appendix. Encouragingly, these additional experimental results demonstrate that our proposed method continues to outperform other baseline methods in terms of the ATE error metrics. This further validates the effectiveness of our approach and strengthens our contributions to the field.

6. Report of CATE and ATE estimates on real data.

Thank you for your question about reporting the CATE and ATE estimates on real-world data.

Regarding CATE estimates, due to the nature of real-world data, which lacks counterfactual outcomes for each sample, it is unfortunately not feasible to calculate CATE as it is defined by the average root mean square difference between the actual and predicted CATE for each sample: ${\epsilon_{PEHE}}=\sqrt{\frac{1}{N}\sum_{i=1}(\tau_i-\hat{\tau}_i)^2}$,

where $\tau_i=y_{i}^{t_{i}=1}-y_{i}^{t_{i}=0}, \hat{\tau_i}= \hat{y}_i^{t_i=1}-\hat{y}_i^{t_i=0}$ are the ground truth CATE and the estimated CATE, respectively.

For the ATE error metric, while the standard definition involves the absolute difference between the true ATE and the predicted ATE: $|Error_{ate}| = |ATE -\hat{ATE}| = \left| \frac{1}{n} \sum_{i=1}^{n}(y_i(1) - y_i(0)) - \sum_{i=1}^{n}(\hat{y}_i(1) - \hat{y}_i(0)) \right|$, the actual calculation requires knowledge of each sample's true counterfactual outcome, which is not available in real-world data.

However, understanding the importance of your inquiry, we approximated the ATE calculation as follows: $ATE = \frac{1}{m}\sum_{i=1}^{m} y_i(1) - \frac{1}{k}\sum_{i=1}^{k} y_i(0)$, where m and k are the number of samples in treated and control group. Then we conducted experiments on real-world data using this approximation. In our study, we focused on our proposed method PoNet, along with CFR and TarNet (excluding others like OLS1, OLS2, TEDVAE, BART, Causal Forest, GANITE, CEVAE due to their significantly larger error values). We recorded the ATE error metrics for these methods every 10 epochs during 100 training epochs:

The results indicate that the ATE error metrics are highly unstable, appearing somewhat random. This instability is likely due to the unavailability of true ATE values in real-world data, suggesting that calculating ATE error metrics without counterfactual outcomes may not be a reliable evaluation metric in real-world settings. We hope this explanation clarifies our approach and findings, and we appreciate your understanding of the inherent limitations when working with real-world data.

1.Legends in figures and tables are not clear enough.

Thank you for pointing this out. We are sorry that the legends in the figures and tables are not clear for reading. In the revised version, we have revised all figures and tables to include comprehensive legends. These will detail all variables and key aspects, ensuring they can be understood independently of the main text. We appreciate your attention to these details.

2. Source code is not provided.

Thank you for your insightful feedback regarding the availability of the source code for our work. We would like to kindly point out that the source code for our model is indeed available and the link has been provided in the supplementary material. Specifically, the link to the code can be found in Appendix Section E, which discusses additional experiments. We understand how crucial it is for the code to be easily discoverable and accessible, and we apologize if its location in the supplementary material was not immediately apparent. To address this, we have put the code link at the beginning of the experiment section in the main body of the paper.

3. The theoretical part could be more enriched.

Thank you for your valuable feedback regarding the theoretical contributions of our paper. We appreciate your suggestion to enrich the theoretical framework.

Firstly, we would like to highlight that our paper already proposes a significant theoretical contribution regarding the identification of causal treatment in the presence of both confounders and post-treatment variables, as detailed in Theorem 1. We have provided extensive proof for this in the appendix.

Regarding the bounds of reconstruction loss, we have thoroughly considered this aspect. The reconstruction loss based on mean square error, a common regularization term in deep learning, serves dual purposes in our work. It ensures learning about collider post-treatment variables during backpropagation and retains the information in the original covariates. However, the challenge lies in the fact that the reconstructed covariates are processed through a nonlinear decoder based on a multi-layer perceptron. This creates a non-convex problem for solving the bound of this reconstruction loss, presenting a complex challenge in optimization.

But we still totally understand your concerns about the theoretical depth, thus we have further enriched our paper from another angle in the revised version. Noting that Theorem 1 indicates that the treatment effect can be identified by recovering $C$ and $Z_m$, we further provide a new theorem:

The joint set of inferred factors for $C$, $Z_m$ is minimally sufficient for the optimal parameters $\theta$ which estimation of unbiased treatment effects needs.

It refers to the inferred confounders and mediation post-treatment underlying factors are minimally sufficient for recovering the true treatment effect's optimal parameters. This theory asserts that the inferred post-treatment variables and confounders are the most parsimonious statistics, yet they retain all necessary information for estimating the optimal parameters required for a sufficient estimation of the true treatment effect. The theory has been updated in Section 2 of the main body and detailed proof of this new theorem is also provided in the appendix C.

4. Inclusion of a method proposed by Li et al. 2022 in the benchmark.

Thank you for your suggestion to include the method proposed by Li et al. 2022 in our benchmark comparisons. We acknowledge the relevance of this method, as it addresses one aspect of post-treatment bias, which is a key focus of our work. we indeed intended to include Li et al. 2022's method in our comparative analysis. However, the source code for their method is not publicly available. The absence of this code posed a barrier to implementing their method accurately for a fair and rigorous comparison in our experiments. we are open to including this comparison in future work, should the source code become available or if we can collaborate with the authors for an accurate implementation. In the meantime, we have made efforts to ensure that our benchmark comparisons are comprehensive and include a range of relevant methods that are available for comparison. We appreciate your understanding in this matter and are grateful for your suggestion.

3. Concerns about DAG,and limitations of the proposed approach

Thank you for your insightful query about graph structure as depicted in Figure 1c. Our study acknowledges and addresses the omnipresence of post-treatment variables in real-world scenarios and the complexities they introduce in causal inference. The proposed DAG in our paper is designed to illustrate a general framework for handling these complexities, but we understand that its applicability might vary based on the specific context and knowledge of the covariates involved.

The question of how many covariates are "needed" for the effective application of the proposed model is also an important consideration. Our experiments, which encompassed synthetic, semi-synthetic, real world datasets with dimensions ranging from tens to thousands of covariates, have demonstrated that PoNet is effective even in scenarios where the number of covariates is relatively small. This effectiveness is particularly notable in comparison to methods that only consider confounders. The strength of PoNet lies in its data-driven decomposition to identify and distinguish between pre-treatment and post-treatment covariates when the pre-post nature of covariates is unknown. We acknowledge, however, that in some extreme scenarios, such as when there is only a single covariate, PoNet might not perform optimally. This limitation arises from PoNet's design, which separates out three distinct types of factors from the covariates. In such extreme cases, our model might not be as effective, but it is important to note that this limitation is not unique to PoNet and applies to other popular causal estimation methods as well. We appreciate your insightful perspective on this matter. It highlights an interesting point about the relationship between the choice of estimation method and the contextual background knowledge of the application scenario. Practitioners should carefully consider the nature of their data and the specific requirements of their research context when choosing a causal estimation method.

Another limitation of this study is our assumption that the mediation and collider post-treatment variables are entirely independent. Our theoretical analysis suggests that to mitigate post-treatment bias, we should condition on mediation post-treatment variables while excluding collider post-treatment variables in the inference policy. However, it is possible for certain post-treatment variables to act as both mediators and colliders simultaneously. This duality introduces the risk of collider post-treatment bias when they are included as conditioning factors in inference, and mediation post-treatment bias when they are not. Consequently, further exploration is warranted in this area. It is crucial to investigate whether there are more theoretically sound strategies to effectively eliminate such biases or to determine the prevalence of dual-type post-treatment variables in real-world settings and carefully select the scenarios for application, thereby offering more insightful guidance to practitioners.

We have updated the above limitation discussion in the updated Appendix. Thank you very much for your valuable comments!

4. Question about the DAG in terms of degenerate distribution

In our DAG, the arrow $X \rightarrow C, Z_c, Z_m$ indicates that the confounders $C$ and post-treatment variables $Z_m, Z_c$ are a subset or a function of the covariates $X$. This does not necessarily imply a deterministic partitioning of $x$ into distinct components $Z_m$, $Z_c$ and $C$. Instead, it suggests that $C$ (and $Z_m$, $Z_c$) can be derived or inferred from $X$. We just use expression forms such as $P(C|X)$ to highlight the distributions of the factors of interest can be inferred from covariate.

But we do understand your confusion and concerns regarding this expression form, in response, we have followed the work of Zhang et al 2021 to provide a more simplified proof of Theorem 1. This re-formulated proof also hinges on Theorem 1 that if we can infer the distributions of $C$, $Z_m$ and $Z_c$, then the causal effect is identifiable under the assumed causal structure. The detailed proof, which we have now updated in the appendix, is as follows:

$P(Y_u|do(T),X) =P(Y|do(T),C,Z_c,Z_m,U)\\ =P(Y|do(T),C,Z_m,U)\\ =P(Y|T,C,Z_m,U)\\ =P(Y_u|T,C,Z_m)$

This proof, although reformulated, employs the same tools and principles as the original expression form, grounded in the causal graph we proposed, the corresponding manipulated graph, and the three rules of the do-calculus. The last term of the derivation is precisely the same as $P(Y_u|T,Z_m,C)$ in the last term of our original formal derivation (See Appendix C), but here the process of inferring Z_c, Z_m and C from X is omitted. We believe this clarified approach more effectively addresses your concern and elucidates how we apply these theoretical tools to support the causal inference capabilities of our model.

1.Some redundant discussion of well-known issues and the connection of Theorem 1

We appreciate your suggestion to reduce the emphasis on the general problem of post-treatment bias, given its established recognition in the field. We agree that a more focused approach would indeed allow us to highlight the unique contributions of our work more effectively. But detailing the two sources of different post-treatment bias in the paper is crucial for the readability of the article and for the reader to understand the contribution of our work. Based on your suggestions, we have tried to shorten the description of the post-treatment bias in the revised version.

Regarding Theorem 1, we acknowledge the need for a clearer connection to the specific problem of post-treatment bias. Our intention with Theorem 1 is to establish that under certain causal structures, the knowledge of the conditional distribution of pure confounders and post-treatment variables given all variables is critical for identifying causal effect. In Theorem 1, we demonstrate a key insight: under the given conditions of covariates $X$ and treatment assignment $T$, the probability distribution of an outcome under an intervention $T$ is determined by the distribution of confounders and mediation post-treatment variables, rather than by collider post-treatment variables. This aligns directly with our analysis presented in Section 2.2. In that section, we discuss the importance of including mediation post-treatment variables in causal effect estimation to capture the indirect effects of treatment on outcome through the mediation variable $Z_m$. Conversely, we argue against conditioning on collider post-treatment variables $Z_c$, as this could introduce additional bias due to unobserved variables. This aligns perfectly with the conclusion of Theorem 1, which underscores the importance of accurately identifying and including the right variables in causal models to avoid post-treatment bias. And this theorem is pivotal in demonstrating how our proposed method effectively handles the complexities introduced by post-treatment variables, thereby mitigating post-treatment bias. Our work with Theorem 1 thus directly addresses the challenge of post-treatment bias by elucidating the specific types of variables that should be considered in the causal modeling process. This theoretical grounding strengthens the basis for our proposed model, PoNet, which is designed to effectively distinguish and utilize the appropriate post-treatment variables for accurate causal inference. Thank you for highlighting the need to more explicitly connect Theorem 1 to the issue of post-treatment bias. Your feedback has provided us with an opportunity to clarify this crucial aspect of our paper. We have clarified it in the updated version.

2.Some distracting aspects, e.g., the discussion of MIMR.

We appreciate your appreciation of the empirical results regarding MIMR, and we agree that a more focused discussion on the specific problem it addresses would enhance the paper's clarity.

The primary purpose of the MIMR, as detailed in Section 3.3 of our paper, is to ensure the accurate separation of confounders and post-treatment variables, which is essential for unbiased treatment effect estimation. In situations where representations of confounders include information from mediation post-treatment variables, controlling for confounders might inadvertently introduce mediation post-treatment bias. Similarly, if mediation post-treatment variables contain confounder information, addressing confounding bias might not be fully effective. if mediation post-treatment variables $Z_m$ contain collider post-treatment information, conditioning on $Z_m$ to predict the outcome can lead to collider post-treatment bias as illustrated in Section 2.2. Precise differentiation between confounders and post-treatment variables is thus critical for reliable treatment effect estimation. To address this challenge, we utilize the MIMR based on kernel density estimation. This non-parametric method fits the distributions of the representations of these variables and measures their independence. By minimizing the mutual information between these variable representations, we effectively separate the underlying factors from each other, thus preventing the inadvertent introduction of bias into the treatment effect estimation. This separation is critical, as it allows for more precise differentiation between confounders and post-treatment variables, ensuring reliable treatment effect estimation. The MIMR thus plays a pivotal role in enhancing the robustness and accuracy of PoNet, our proposed model, by ensuring the independence of critical variables in the causal inference process. We hope this explanation clarifies the specific problem that MIMR addresses and its importance in the context of our work. We have emphasized this aspect more clearly in the revised version of our paper.

Dear Reviewer,

I hope this message finds you well. I am writing to inquire about the response we recently submitted addressing your concerns regarding our manuscript. We have not yet received your feedback, and we are eager to know if our responses and revisions have adequately addressed your concerns.

We deeply appreciate the score of 3 you initially assigned, which we believe reflects your honest and critical evaluation of our work. In light of this, we have taken your feedback very seriously. Following the insights provided by you and other reviewers, we have conducted substantial additional experiments and developed new theoretical analyses. These efforts have led to solid modifications and significant improvements in our manuscript. Understanding the rigorous standards of the review process, we have worked diligently to enhance the quality and impact of our paper. We believe that these revisions have substantially addressed the initial concerns and have enriched the contribution of our work.

Considering the extent of these revisions and enhancements, we kindly request you to reconsider the evaluation of our paper. We would be grateful if you could review the updated version of our manuscript and provide your valuable assessment.

We understand the demands on your time and appreciate your dedication to maintaining the high standards of the conference. Your expertise and feedback are crucial to us, and we sincerely hope that our revised submission aligns more closely with these standards.

Thank you very much for your time and consideration. We look forward to your thoughts and are hopeful for a positive re-evaluation.

Best regards,

Authors