Thank you for your thoughtful review and for acknowledging the strengths of our paper, and we hope our feedbacks below will help address your concerns.

Weakness 1: Necessity of enforcing $\hat{Y} \perp A \mid Y$ instead of $\hat{Y} \perp A_k \mid Y$ for each $k$ individually

Thank you for asking this important question "is it sufficient to just consider $\hat Y \perp A_k \mid Y$ for all $k$ instead of $\hat Y \perp (A_1,...,A_k) \mid Y$?". In fact, the negative answer to this question has been given in many previous literature in the field of fair ML (Kearns et al.[1], Hébert-Johnson et al.[2] and Kim et al.[3]). As highlighted by Kearns et al.[1], focusing only on marginal fairness (i.e., enforcing fairness for each sensitive attribute separately) can lead to a situation called fairness gerrymandering. In this scenario, a model appears fair when considering each attribute individually but exhibits significant unfairness for certain subgroups defined by combinations of attributes. We refer to a simple example provided by Kearns et al.[1] to illustrate why it's not enough to just enforce a model to be fair towards each sensitive attribute separately: Imagine a setting with two binary features, corresponding to race (say black and white) and gender (say male and female), both of which are distributed independently and uniformly at random in a population. Consider a classifier that labels an example positive if and only if it corresponds to a black man, or a white woman. Then the classifier will appear to be equitable when one considers either protected attribute alone, in the sense that it labels both men and women as positive 50% of the time, and labels both black and white individuals as positive 50% of the time. But if one looks at any conjunction of the two attributes (such as black women), then it is apparent that the classifier maximally violates the statistical parity fairness constraint. Thus, considering the joint $A_1, ..., A_k$ and take their dependencies into account when designing a fair algorithm is definitely necessary in this field.

We appreciate the reviewer for raising this point, and we have included this discussion into our revised version.

Weakness 2: Contributions of FairICP and its difference from FDL

While we build upon the adversarial framework introduced by Romano et al.[4], our method differs significantly in the targeted problem and in the way we generate the synthetic sensitive attributes and enforce fairness.

• Targeting a more general but challenging objective: our method is specifically designed to better handle complex sensitive attributes that can be multi-dimensional and span continuous, categorical, or mixed-type. In contrast, FDL does not discuss or provide solutions for the challenges associated with estimating the conditional density $q(A∣Y)$ when $A$ is complex, making FDL not suitable for this objective, which is important in the community of fair ML as we discussed in the introduction and in the response to Weakness 1.

• Theoretical analysis and empirical demonstration of ICP: we provide a theoretical justification for ICP sampling strategy (Theorem 2.1), and empirically demonstrate a more superior performance in terms of the quality of permutation than other conditional randomization/permutation methods (Section 3). We view this as an advance in statistical method and can be applied in various statistical and machine learning contexts where conditional independence is of interest.

• Introduction of a general non-parametric measure and hypothesis testing procedure to the field of fair ML: our "Hypothesis Test for Equalized Odds with ICP" introduce a fairly adaptable and non-parametric measure (KPC) to gauge the violation of equalized odds for all models and for all kinds of sensitive attribution, and we combine it with ICP to develop a formal hypothesis testing procedure. These metrics perform a direct comparison with A, in contrast to the Fair Dummies test from Romano et al.[4], the novelty of which is also acknowledged by other reviewers.

Again, we sincerely appreciate the reviewer for providing valuable feedbacks and questions, and we hope our responses above could help address your concerns. We are also willing to answer any further questions you may have, thank you!

Question 2: "I understand the advantages of using Eq. 7 over Eq. 6, while I am not certain if Eq. 7 is a perfect alternative. Is there a condition for applying ICP here?"

Thank you for raising this point. Yes, Eq. (7) represents the sampling algorithm of Inverse Conditional Permutation (ICP), and Eq. (6) represents that of Conditional Permutation (CP). Both ICP and CP can be applied without strict conditions; however, their effectiveness depends on the quality of the estimated conditional densities (section 2.2 and our simulation comparing ICP and CP).

There is no specific condition limiting the usage of ICP. As with permutation-based methods, a sufficiently large sample size is needed to ensure that the asymptotic properties hold, as indicated by our theoretical results (Theorem 2.1 in the paper). We demonstrate in our theorem that the convergence rate is fast with respect to the sample size $n$, so applying ICP does not introduce significant bottlenecks compared to the training of the prediction model itself.

Question 3: "If I have corrected understand the proposed method, the trained model would be only fair in terms of EOd on observed sensitive attributes. I mean if a new sensitive attribute is specified during testing, it cannot be handled."

Thank you for pointing this out. Like all the in-processing methods (see background and related work in our section 1 ), FairICP is designed to enforce equalized odds during training. If a new sensitive attribute is introduced during testing that was not accounted for during training, all in-processing methods may not satisfy equalized odds with respect to this new attribute (unless the model is retrained or updated). Handling fairness with respect to new or unseen sensitive attributes during testing would fall under post-processing fair ML methods.

With this being said, our proposed testing procedure (including the usage of KPC) can be used to detect violations of equalized odds with respect to any new sensitive attributes for any models. This allows people to assess the fairness of the model concerning additional attributes and take appropriate actions if necessary.

Question 4: "Can you explain the meaning of Recap on line 180?"

In the section starting at line 180, we provide a brief recap of the Conditional Permutation (CP) sampling strategy and describe the potential use of CP for encouraging equalized odds. This builds the foundation for our ICP procedure for equalized odds learning. By understanding CP and its applications, we can better explain the motivation behind our Inverse Conditional Permutation (ICP) method. The ICP builds upon the sampling algorithm designed for CP but alleviates the challenge in handling complex sensitive attributes by focusing on estimating an easier density $Y | A$ instead of $A | Y$. Overall, this recap 1) sets the stage for introducing ICP, highlighting the differences and advantages of our proposed method over CP in the context of enforcing equalized odds with complex sensitive attributes, and at the same time, 2) makes sure that proper credit is given to previous work and avoids overclaimed contribution of ICP.

We are grateful for your valuable feedback and we hope our answers will address your concerns. Please feel free to let us know if you have any other questions.

References

[1] Tang, Z., & Zhang, K. (2022, June). Attainability and optimality: The equalized odds fairness revisited. In Conference on Causal Learning and Reasoning (pp. 754-786). PMLR.

[2] Romano, Yaniv, Stephen Bates, and Emmanuel Candes. "Achieving equalized odds by resampling sensitive attributes." Advances in neural information processing systems 33 (2020): 361-371.

We appreciate the constructive feedbacks by the reviewer, here are our responses:

Weakness 1: "They used complex sensitive attributes in the paper. But according to the content, it only refers to multiple independent attributes. The correlation of attributes are not touched. Please clarify this point."

We apologize for any confusion on this point. In our work, "complex sensitive attributes" refers to sensitive attributes that are multi-dimensional and can be of arbitrary types (continuous, categorical, or mixed). Our method does not assume that these attributes are independent. In fact, FairICP can handle correlated sensitive attributes effectively.

In our simulation (see section 3.1.1 for simulation setup), we generated correlated multi-dimensional sensitive attributes $A$ by generating a random covariance matrix. In real-data experiments, we consider cases where the sensitive attributes are indeed correlated. For example, in the Communities and Crime dataset, the percentages of different minority races (African American, Hispanic, Asian) within a community are inherently related, as they sum up to a certain proportion of the population.

We thank the reviewer for pointing this out and we have clarified this point in the revised version of the paper.

Weakness 2: "I felt confused why this work only focuses on EOd metric when I started reading tit. But it turns out that the ICP might be only appropriate for this setting. Is this a limitation of ICP?"

Thank you for pointing this out. Our focus on the equalized odds (EOd) metric is intentional, as enforcing equalized odds requires ensuring conditional independence. This conditional nature makes it a more challenging fairness criterion compared to unconditional ones like demographic parity (Tang et al.[1]).

While ICP is well-suited for enforcing equalized odds due to its ability to avoid the challenging density estimation of $A$ given $Y$, for unconditional fairness notions such as demographic parity, we can utilize unconditional permutation of $A$ which is a special case of FairICP without conditioning.

Weakness 3: "Why KPC and loss trade-off is highlighted in experiments? Typically, ACC/Error and EOd are expected to be shown."

Thank you for raising this point. In our experiments, we highlight the trade-off between prediction loss (MSE in regression or misclassification rate in classification) and the violation of equalized odds measured by the Kernel Partial Correlation (KPC) coefficient, which is exactly the "ACC/Error and EOd" trade-off as the reviewer pointed out. We chose KPC to measure Eod because it is a rigorous, non-parametric measure of conditional independence that quantifies the degree of violation of equalized odds. KPC is applicable to both regression and classification tasks and can handle sensitive attributes of arbitrary types and dimensions.

Weakness 4: "$\tilde A$ should be "generated" and labeled as "fake" in Eq. 3. Then in the last sentence of caption in Fig. 1, it should be regularizing $\tilde A$ towards $A$."

Thank you for pointing this out. $\tilde A$ is consistently used to represent generated sensitive attributes throughout our paper (and also in FDL paper Romano et al.[2]). In our work, $\tilde A$ is guaranteed to achieve equalized odds asymptotically (i.e., $\hat Y \perp \tilde A | Y$), thus we describe the training mechanism as "regularizing $(\hat Y, A, Y)$ towards $(\hat Y, \tilde A, Y)$" to achieve this desired property. We have now also highlighted that $\tilde{A}$ is a “fake” copy of A before Eq. 3 and in the legend of Figure 1 to remind readers of its meaning.

Question 1: "Regarding FairICP’s flexibility, are you referring to that the proposed technique can be used for both classification and regression?"

The flexibility of FairICP refers to two main aspects:

1. Handling complex sensitive attributes: FairICP is designed to handle sensitive attributes that are multi-dimensional and can be of arbitrary types, including continuous (see Communities & Crimes dataset results), categorical (Adult / COMPAS dataset), or mixed (ACS Income dataset). This includes multiple sensitive attributes simultaneously, regardless of whether they are independent or correlated. This aspect is the main emphasis of FairICP as handling such complex sensitive attributes efficiently is challenging task unresolved in current literature.

2. Efficacy to both classification and regression tasks: Our framework is applicable to both regression and classification problems. By selecting an appropriate prediction loss function, FairICP can enforce equalized odds fairness in either setting. This makes it versatile for a wide range of machine learning tasks. We demonstrate this versatility in our experiments, by showing regression task on Communities & Crimes dataset and classification tasks on Adult/COMPAS/ACS Income dataset.

Thank you for your follow-up question!

The permutation remains meaningful even when the true sensitive attribute is directly correlated with $X$. Consider the most extreme case where $A$ is actually a feature in $X$ (e.g., $A$ = $X_1$). The generated fake dataset still only replaces $A$ with the fake sensitive attribute without using such information - that is, ($\tilde A$, $X$, $Y$) = ($\tilde X_1$, $X$, $Y$) where $\tilde X_1$ is generated based on the conditional distribution of $X_1 | Y$. This breaks the dependence structure between $\tilde A$ and $X$. In this scenario, we will encourage the trained model to only use $X_1$ when its inclusion contributes to response prediction fairly across samples with different $X_1$ values. In fact, it is common in real-world datasets for $A$ to be naturally correlated with $X$, thus influencing $Y$ through $X$ (Kusner et al. [1]), and our method is designed to handle such situations effectively.

Regarding the image-type dataset, for example, CelebA (which has been used a lot in the field of algo fairness, e.g., Creager et al. [2]), there are 40 features that can serve as $A$ (e.g., OvalFace, HeavyMakeup, Male, etc.). Thus, the implementation is similar to that of the tabular dataset, with which our framework is quite compatible.

We hope this clarifies your concern and please don't hesitate to let us know if you have any additional questions.

Reference

[1] Kusner, Matt J., et al. "Counterfactual fairness." Advances in neural information processing systems 30 (2017).

[2] Creager, Elliot, et al. "Flexibly fair representation learning by disentanglement." International conference on machine learning. PMLR, 2019.

Thank you for your thoughtful and detailed review of our paper. We appreciate your recognition of the strengths of our work, including the theoretical contributions and the novelty of our hypothesis testing procedure, and we are committed to addressing each of your concerns to improve our paper.

1. Weakness 1: lack of justification of the difficulty of sampling from $q(A | Y)$

   Thank you for raising this point. Estimating the conditional distribution $q(A∣Y)$, or even the density $q(A)$, is known to be challenging as the dimension of $A$ increases. Non-parametric density estimation methods, such as kernel density estimation, suffer significantly in high dimensions due to the well-known phenomenon of curse of dimensionality. The required sample size for accurate estimation scales exponentially with the dimension $p$ (Scott [1]). Also, when $A$ includes multiple correlated or even mixed-type variables, modeling the joint conditional distribution $q(A∣Y)$ becomes complex. For categorical variables, combining categories to model dependencies leads to exponentially decreasing amount data in each category, making estimation unreliable. We have included this additional discussion on this issue in line 171-173.

   Empirically, we compare the performance of ICP and other methods that are based on estimating $q(A | Y)$ in our simulation studies, where we systematically investigate the permutation quality by varying dimensionality of $A$ and noise levels.

2. Weakness 2: Overly detailed derivation in Theorem 2.1

   Thank you for this feedback. We have now rephrased our arguments to make it more concise.

   • In task 1, after establishing the equivalence between ICP and CP given the correct conditional density estimation $Y|A$, we used 3 lines to conclude the proof of task 1.
   • In task 2, we have now removed some intermediate steps to avoid being overly detailed. We still separate Task 2 out as this implied the difference between conditional independence and the conditional independence after conditioning on the observed sensitive attribute set.

3. Weakness 3: Code availability

   We apologize for not including access to our code in the initial submission. We have now provided the relevant code in supplementray materials for you to check.

4. Weakness 4: Use of antiquated fairness datasets

   Thank you for bringing this to our attention. We agree that using more contemporary and realistic datasets is crucial for advancing fairness research.

   • For ACS Income data, we have run a new experiment with sex/race/age (mixed types) as sensitive attributes, which hasn't been explored in the previous literature to the best of our knowledge. In this case, we compare our FairICP with HGR (which is the only baseline model we can think of to handle this type of sensitive attribute) and again demonstrate a better performance.

   • For COMPAS data, we have briefly discussed its limitations in our initial submission (line 483-485).

5. Weakness 5: Presentation of empirical results and metrics

   Thank you for pointing this out.

   • More standard metrics, for classification task with discrete sensitive attributes (Adult and COMPAS), we report the DEO (differences of observed equalized odds) in the table and we also add the Accuracy-DEO trade-off curve in the appendix.

   • Explanation on Adult results: In Table 1, we reported the results using the second-largest hyperparameter values for FairICP and HGR to achieve a comparable level of KPC to Reduction at its largest hyper-parameter value (see Figure 4). However, with this choice of penalty, the differences in prediction loss between FairICP and FDL were not particularly evident. When we instead use the largest hyperparameters for both FairICP and HGR (resulting in KPC values smaller than that for Reduction, but still similar to each other), the differences become more noticeable: FairICP = 0.159(0.004), HGR = 0.162(0.004), which still yield comparable KPCs. We have updated the table 1 with more experiments and more metrics.

6. Clarification on theorem 2.4

   We apologize for the lack of clarity in our original submission regarding the term "optimality." We have revised our submission to provide precise definitions. Specifically, we now clarify that if a solution achieves the minimum possible value with ICP-generated synthetic copies $\tilde{A}$, as stated in Theorem 2.4, this solution is both an optimal predictor whose prediction loss is the conditional entropy H(Y | X) and asymptotically fair as stated in the revised Theorem 2.1.

Once again, we thank you for your thorough review and constructive feedback. We hope that our responses address your concerns and we are willing to address any further concerns you may have.

References

[1] Scott, David W. Multivariate density estimation: theory, practice, and visualization. John Wiley & Sons, 2015.

Dear Reviewer MX2Y,

We want to clarify that this paper's primary goal is to introduce ICP as a flexible framework for encouraging equalized odds, comparing different strategies using consistent prediction model architectures. While combining gradient boosting with these strategies is valuable, such implementations merit dedicated research papers (particularly adversarial learning-based ones like FDL and FairICP). For instance, FairGBM demonstrates this by integrating the reduction approach with gradient boosting for categorical sensitive attributes.

While developing specific machine learning tools under the FairICP framework is an important direction for future work, we believe these case-specific implementations should be separate from the current methodology paper. We revised our discussion to discuss this:

"FairICP offers a flexible framework for encouraging equalized odds with multi-dimensional sensitive attributes of various types. It can be combined with other prediction models beyond the linear model and neural network structures explored in this paper for a potential increase in performance. For example, FairGBM has achieved state-of-art performance in CSIncome and ACSPublicCoverage datasets with categorical sensitive attributes when integrating gradient boosting with the idea of Reduction [1,2]. We defer such additional realizations of FairICP framework to other prediction model architectures as a future direction."

We respectfully request you to consider whether gradient boosting implementation should be viewed as future work rather than essential to the current paper, given that such implementations require substantial engineering work, as shown by FairGBM. Thank you again for your feedback!

[1] Cruz, A.F., Belém, C., Jesus, S., Bravo, J., Saleiro, P. and Bizarro, P., 2022. Fairgbm: Gradient boosting with fairness constraints. arXiv preprint arXiv:2209.07850.

[2] Cruz AF, Hardt M. Unprocessing seven years of algorithmic fairness. arXiv preprint arXiv:2306.07261. 2023 Jun 12.

6. Misstatement 1: "Proposition 2.5 is only valid when the true conditional density is known."

   Thank you for pointing this out. You're right to assume correct density estimation, just as in FDL paper (Romano et al.[4]) and Holdout Randomization Test paper (Tansey et al.[5]). We have assumed the correctness of the conditional density estimation Y|A throughout Section 2, which has now been clarified in our revision.

7. Misstatement 2: "The derivation in Lines 678-694 is meaningless when the sensitive features are real-valued and admit density, as the probabilities in Eqs. (10) and (11) are always zero. In Eq. (12), $P(Y=y, S(A)=S)=0$ for real-valued $Y$ and/or $S$, resulting in zero division. Similarly, in Line 729, $P(Y=t^3)=0$ for real-valued $Y$"

   When it is continuous, it means density. For example, for continuous $y$ it means $P(Y=y, S(A)=S)=p_Y(y)P(S(A)=S|Y=y)$, etc. Similarly, when we use the density notation p(.), it represents the point mass for discrete distributions. We have used them interchangeably throughout the paper, and have clarified it in our revision (Appendix A. PROOFS).

8. Misstatement 3: "Definition of $S$-induced empirical CDF"

   Thank you for pointing this out. As illustrated in line 764-768, let $S$ be a set $n$ samples $A$ generated conditioned on $Y=t^3$, the empirical CDF $\hat F_{t^3}^S(a) = \frac{1}{n} \sum_{i=1}^n \mathbb{I}_{A_i \leq a}$. The probability of interest:

   $$\mathbb{P}\left(A_1 \leq a \mid \mathbf{Y}=\mathbf{t}^3, S(\mathbf{A})=S\right) = \frac{(n-1)!\sum_{i=1}^n I_{A_i \leq a}}{n!} = \hat F_{t^3}^S(a)$$

   by the exchangeability of $A_1,...A_n$ given $Y$ and $S$, which justifies the usage of the theorem in Naaman [6]. We have included it in our revised version.

9. Misstatement 4: "Proof of attainability of lower bound in Theorem 2.4"

   Thank you for pointing this out. The proof that the equality of lower bound in line 792 can be achieved by the minimizer (assume it exists) is in line 795-800. Similar results and proofs can be also seen in Goodfellow et al.[7] and Louppe et al.[8].

10. FDL on COMPAS and Adult

    We agree with the reviewer that a comparison with the baseline method is important. However, the FDL does not support the generation of multiple category/mixed $A$.

11. MAF for section 3.1.1 result

    Thank you for pointing this out. As stated in line 377, this result is in the Appendix C.2.

Again, we sincerely appreciate your constructive feedbacks and we hope our answers above will help address your concerns. We are also willing to answer any further questions you may have, thank you!

References

[1] Narasimhan, Harikrishna, et al. "Pairwise fairness for ranking and regression." Proceedings of the AAAI Conference on Artificial Intelligence. Vol. 34. No. 04. 2020.

[2] Sugiyama, M., Suzuki, T., Nakajima, S., Kashima, H., von Bünau, P., and Kawanabe, M. (2008). Direct importance estimation for covariate shift adaptation. Annals of the Institute of Statistical Mathematics, 60(4):699–746.

[3] Menon, A. and Ong, C. S. (2016). Linking losses for density ratio and class-probability estimation. In International Conference on Machine Learning, pages 304–313. PMLR

[4] Romano, Yaniv, Stephen Bates, and Emmanuel Candes. "Achieving equalized odds by resampling sensitive attributes." Advances in neural information processing systems 33 (2020): 361-371.

[5] Tansey, Wesley, et al. "The holdout randomization test for feature selection in black box models." Journal of Computational and Graphical Statistics 31.1 (2022): 151-162.

[6] Naaman, Michael. "On the tight constant in the multivariate Dvoretzky–Kiefer–Wolfowitz inequality." Statistics & Probability Letters 173 (2021): 109088.

[7] Goodfellow, Ian, et al. "Generative adversarial nets." Advances in neural information processing systems 27 (2014).

[8] Louppe, Gilles, Michael Kagan, and Kyle Cranmer. "Learning to pivot with adversarial networks." Advances in neural information processing systems 30 (2017).

[9] Scott, David W. Multivariate density estimation: theory, practice, and visualization. John Wiley & Sons, 2015.

We appreciate your thorough review and constructive feedback on our paper, here are our responses:

1. Weakness 1: "However, the paper does not clearly define what "complex" means in this context, leaving the reader with unanswered questions about the method's applicability."

   Thank you for pointing this out. In our paper, "complex sensitive attributes" refer to attributes that are multi-dimensional and can be of arbitrary types, including continuous, categorical, or mixed. This includes multiple sensitive attributes simultaneously, regardless of whether they are independent or correlated. This was originally described as we discussed existing methods for equalized odds learning, and we have now revised our manuscript to highlighted it before such discussions (Section 1: background and related work).

2. Weakness 2: "Capability to handle complex sensitive features is not novel."

   Thank you for raising this point. While existing methods like HGR and FDL can be applied to handle to settings with multi-dimensional continuous sensitive attributes, they in fact face significant practical challenges: HGR requires estimating maximal correlations conditioned on $Y$, when it comes to multi-dimensional $A$, it requires a non-trivial extension on the HGR coefficient and not discussed in the original paper (also see our footnote in line 539). We have demonstrated that HGR performed worse compared to FairICP with multi-dimensional sensitive attributes (see Table 1 and Figure 4). Similarly, FDL relies on estimating the conditional density $q(A | Y)$, which is difficult when $A$ is multi-dimensional, e.g., it is well-known that multi-dimensional density is difficult to estimate and requires sample size to scale exponentially with the dimension p for accurate estimation (see Scott [9], in fact, the multi-dimensional cases are also not discussed in the FDL paper, including the density estimation part). We have also had detailed comparisons comparing FDL and FairICP and the gain of using estimated density $p(Y|A)$ over $p(A|Y)$ (see Figure 2, 3 and 4).

Additionally challenges exist when A is of categorical or mixed types with potential dependence structure among different sensitive attributes. For example, in the setting with multiple categorical sensitive attributes, the native approach of considering all possible categories based on the concatenated categorical attributes often leads to each unique category having very few samples left (exponentially decreasing). All these factors contribute to the difficulties when it comes to multiple sensitive attributes.

3. Weakness 3: "literature Narasimhan et al.[1]"

   Thank you for bringing this work to our attention. This literature introduces a new fairness notion based on pairwise comparison of two individuals for a single sensitive attribute, which generalizes the notion of equalized opportunity and is specifically designed for ranking and regression settings. However, these fairness definitions and goals are different from our work, which focus on achieving equalized odds for multiple sensitive attributes in classification/regression setting.

4. Weakness 4: "Need to discuss the efficacy of the trade-off between accuracy and fairness"

   We agree with the reviewer that emphasizing the efficacy of the trade-off between accuracy and fairness is crucial. In our paper, we have provided empirical evidence demonstrating this:

   1. Simulation Studies: In Section 3.1, we present simulations comparing FairICP with existing methods like FDL under varying complexities of sensitive attributes, which was highlighted in the simulation because it served as an ablation study investigating the contribution of ICP. The results show that as the dimensionality and complexity of $A$ increase, FairICP maintains better accuracy-fairness trade-offs.

   2. Real-World Experiments: In Section 3.2, we apply FairICP to various datasets and show a better trade-off between accuracy (Loss) and fairness (KPC/Power).

5. Weakness 5: "They claim that their method, which utilizes the conditional density estimation of $Y|A$, achieves a more efficient trade-off than existing methods employing the conditional density estimation of $A|Y$. However, the paper lacks a logical explanation to support this claim."

   Thank you for raising this point. In addition to the discussion and simulation on the efficiency comparing ICP and other methods (section 3.1), we would like to point out that, our idea of shifting the burden from estimating $X|Y$ to $Y|X$ is reminiscent of the very well-known strategy for density ratio estimation. An important contribution in this classical field is by Sugiyama et al.[2] and Menon and Ong [3] to build a classifier for estimating $P(Y=1|X)$ for learning $p(X|Y=1)/p(X|Y=0)$, which avoids the challenges of estimating a potential high-dimensional density $p(X)$.

Dear Reviewer mGqp,

We wish to further clarify the points you used:

1. " FairCP should exhibit better efficacy in the trade-off on, for example, the Crime (all races) dataset, since the real value ($Y$) is more complex than multiple binary values ($A$)."

   Reply: We want to clarify that we do not claim FairCP exhibits better efficiency trade-offs. FairCP is a new procedure we designed as a middle ground between FDL and FairICP to compare two conditional permutation approaches. It helps illustrate that FairICP's advantage over FDL stems from using conditional distribution estimation of Y|A, not merely due to permutation. FairCP, which uses conditional permutation based on A|Y like FDL, performs worse than FairICP as shown in our simulations.

2. "The authors argue that $\hat\theta_f$ is the minimizer of both $\mathcal L_f$ and $JSD(p_{\hat{Y}AY} || p_{\hat{Y}\tilde{A}Y})$. However, Menon et al. showed that the loss minimizer under the constraint of strict equalized odds, namely when $JSD(p_{\hat{Y}AY}|| p_{\hat{Y}\tilde{A}Y})=0$, can differ from the loss minimizer without the constraint, which contradicts the aforementioned argument. "

   Reply: We want to clarify the validity of Theorem 4.1. The theorem holds because we assume the existence of a solution achieving the minimax loss $(1-\mu)H(Y|X)-\mu \log 4$ (see proof). Under this assumption, the minimizer of the target equation simultaneously achieves both loss minimization and $JSD(p_{\hat{Y}AY}|| p_{\hat{Y}\tilde{A}Y})=0$. We hope this addresses your concern about correctness.