Fairness on Principal Stratum: A New Perspective on Counterfactual Fairness

Thank you for your valuable feedback and the time dedicated to reviewing our work. We address your concerns and questions as follows.

Can the authors demonstrate the proposed ideas on more datasets?

Thank you for pointing this issue! We follow your suggestion to add extensive experiments comparing our method to more baselines on two new datasets: Law and UCI Adult.

• Key observation 1: Our method improves PCF more significantly compared to the original CF metric, in which PCF only focuses $Y(0)≠Y(1)$ but CF focuses on all individuals.
• Key observation 2: Existing CF methods wouldn’t perform better than the proposed approach on the PCF metric.
• Key observation 3: Our post-processing approach exhibits very competitive performance in terms of the trade-off between fairness and accuracy -- our PCF results are the best with only a slight decrease in accuracy.

In addition, we find it's meaningful to add experiments to analyze the power of our proposed test -- "what is the likelihood that an algorithm violating PCF can pass this test (also known as sensitivity)?" on the above two new datasets. The results are shown as below.

The above experimental results align with our theoretical claims for our proposed PCF test in Sec. 4.1, i.e., our PCF test is necessary so that the false positive rate is 0. We also empirically show that our PCF test has a relatively low false negative rate.

In practice, it can be challenging to correctly infer causal relationships from data.

Thank you for raising this concern. We would like to clarify that our method for imposing PCF does not require causal discovery. Considering there is no DAG in real-word datasets, we implement causal discovery to first obtain CPDAG, then sample a DAG as the ground truth for simulating the counterfactuals.

Note that our proposed method does not require a known DAG (or even a CPDAG). Instead, the only assumption we make is the ignorability assumption in line 232, i.e., $A \perp(Y(1), Y(0), D(1), D(0)) \mid X$, meaning that there is no unobserved confounders. We would also like to remark that it is natural to extend our approach to further relax this assumption, such as using sensitivity analysis [7] in causal inference literature. We leave this for future work, considering the orthogonality of these two issues.

Lastly, motivated by the reviewer that obtaining an accurate DAG from observational data is challenging, we note that recent studies have been focused on how to achieve CF with partial DAGs [8, 9]. It would be useful to incorporate these works, but it should still be noted that our approach does not try to infer causal relationships from data.

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Please let us know if you have further questions -- thank you so much!

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References

[1] Kusner, Matt J., et al. Counterfactual fairness. NeurIPS, 2017.

[2] Zuo, Zhiqun, et al. Counterfactually fair representation. NeurIPS, 2023.

[3] Chiappa, Silvia. Path-specific counterfactual fairness. AAAI, 2019.

[4] Imai, Kosuke, and Zhichao Jiang. Principal fairness for human and algorithmic decision-making. Statistical Science, 2023.

[5] Plečko, Drago, et al. fairadapt: Causal reasoning for fair data preprocessing. Journal of Statistical Software, 2024.

[6] Kim, Hyemi, et al. Counterfactual fairness with disentangled causal effect variational autoencoder. AAAI, 2021.

[7] Fawkes, Jake, et al. The Fragility of Fairness: Causal Sensitivity Analysis for Fair Machine Learning. NeurIPS, 2024.

[8] Zuo, Aoqi, et al. Counterfactual fairness with partially known causal graph. NeurIPS, 2022.

[9] Li, Haoxuan, et al. A Local Method for Satisfying Interventional Fairness with Partially Known Causal Graphs. NeurIPS, 2024.

Can you come up with a simple SCM describing a scenario where PCF and CF give different answers? Ideally, where PCF gives the more intuitive result.

Thank you for the constructive suggestion to help us improve the readability of our paper!

First, we define PCF within the SCM framework, which is equivalent to the potential outcome framework used in our original manuscript.

• For notations, $A$, $X$, and $Y$ are defined the same as in CF, and $D$ denotes the decision-making from $A$, $X$, and $Y$, which broadens the $\hat Y$ in CF;
• In SCM, $Y=f_Y(A, X, \epsilon_Y)$ and $D=f_D(A, X, Y, \epsilon_D)$, same as $Y$ and $\hat Y$ in CF;
• PCF requires $P(D_{A←a}(U)=y\mid X=x, A=a)=P(D_{A←a'}(U)=y\mid X=x, A=a)$ only for individuals with $Y_{A←a}(U)=Y_{A←a'}(U)$, i.e., $A$ has no effect on $Y$, whereas CF requires to be satisfied for all individuals;
• The main challenges are how to identify individuals with $Y_{A←a}(U)=Y_{A←a'}(U)$ (see Judea Pearl’s "Principal Stratification – A Goal or a Tool?" for more details within SCM, especially Fig. 1 and Table 1), and how to enforce CF on these individuals, instead of all individuals.

Next, we follow the reviewer's suggestion to provide a very simple SCM satisfying PCF but violating original CF.

• Consider $\epsilon_Y\sim \operatorname{Bern}(0.5), Y=A+(1-A)\epsilon_Y$, and $D=Y$ ($D$ is perfect prediction of $Y$);
• In this way, when setting $A←1$, we always have $D_{A←1}(U)=1$ and $Y_{A←1}(U)=1$, that is, $P(D(1)=1, Y(1)=1)=1$;
• When setting $A←0$, we have $D=Y=\epsilon_Y\sim \operatorname{Bern}(0.5)$, that is, $P(D(0)=0, Y(0)=0)=0.5$ and $P(D(0)=1, Y(0)=1)=0.5$;
• For the joint distribution, we thus have $P(D(0)=0, D(1)=1, Y(0)=0, Y(1)=1)=0.5$ (violating CF) and $P(D(0)=1, D(1)=1, Y(0)=1, Y(1)=1)=0.5$ (satisfying CF);
• The former half violates CF, due to $D(0)=0\neq D(1)=1$, but PCF doesn't care these individuals due to $Y(0)=0\neq Y(1)=1$. While the latter half satisfies CF, due to $D(0)=D(1)=1$;
• As a result, this SCM satisfies PCF defined on individuals $Y(0)=Y(1)$, but violates the original CF defined on all individuals;
• To further enforce CF, the learned predictor $D$ need to change its predictions on the $(D(0)=0, D(1)=1, Y(0)=0, Y(1)=1)$ individuals, which will inevitably sacrifice the accuracy of the predictor due to the updated CF predictions $D'\neq Y$ on these individuals (recap that $D=Y$ meets PCF).

Assume we can exclude exogenous confounders from the set of endogenous variables {A, D, Y, X}. Can you give examples what are the general graphical conditions for which PCF is distinct from CF? Ideally, specfiying a DAG.

• The above specified DAG provides a valid example for which PCF is distinct from CF, then we discuss the general graphical conditions for which PCF is distinct from CF.
• Intuitively, as the reviewer noted, the most interesting cases (that do not reduce to standard counterfactual fairness) are where for some sub-population A does not cause Y and for the rest sub-population A does cause Y.
• As an extreme case, if A never cause Y, then we always have $Y_{A←a}(U)=Y_{A←a'}(U)$, making PCF degenerates to CF. Instead, if A cause Y for all individuals, then PCF degenerates to no fairness constraint.

More discussion on the ignorability assumption

• The reviewer is correct that we assume X contains all confounders between A and {D, Y}, but we don't assume that D is conditioned on all X.
• The ignorability assumption only assumes that all confounders are observed, instead of assuming all confounders are conditioned by D and making the backdoor path be blocked.
• We would like to kindly remark that it's natural to extend our approach to avoid the usage of the ignorability assumption, such as using sensitivity analysis [1] in causal inference literature. We leave this for future work, considering the orthogonality of these two issues.
• We thank the reviewer for pointing out this issue and referring us many insightful references, we will definitely cite and discuss them in our final version.

Interpretation of Theorem 2

• Theorem 2 shows the proposed DR estimator can unbiasedly estimate $P(D(a)=d, Y(a)=y \mid X=x)$ in Sec. 4.1 with large samples.

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We are eager to hear your feedback. We’d deeply appreciate it if you could let us know whether your concerns have been addressed.

Validation is limited to the OULAD dataset

• We kindly ask the reviewer to refer to the rebuttal we provide to Reviewer M8WL, in which:
  • We add extensive experiments comparing our method to more baselines on two new datasets: Law and UCI Adult;
  • We also add experiments to analyze the power of our proposed test -- "what is the likelihood that an algorithm violating PCF can pass this test (also known as sensitivity)?" on the above two new datasets.

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Reference

[1] Fawkes, Jake, et al. The Fragility of Fairness: Causal Sensitivity Analysis for Fair Machine Learning. NeurIPS, 2024.

Thank you for your encouraging words and valuable feedback! Below, we address your questions and indicate the changes we’ve made thanks to your suggestion.

Unfortunately, no comparison to alternative fairness definitions are considered. Only one real-world dataset (OULAD) is used. Lack of reporting accuracy metrics before and after fairness adjustments (trade-offs).

Thank you for pointing this issue! We follow your suggestion to add extensive experiments comparing our method to more baselines on two new datasets: Law and UCI Adult.

• Key observation 1: Our method improves PCF more significantly compared to the original CF metric, in which PCF only focuses $Y(0)≠Y(1)$ but CF focuses on all individuals.
• Key observation 2: Existing CF methods wouldn’t perform better than the proposed approach on the PCF metric.
• Key observation 3: Our post-processing approach exhibits very competitive performance in terms of the trade-off between fairness and accuracy -- our PCF results are the best with only a slight decrease in accuracy.

In Section 4.1, the fairness violations may not always be detected accurately (false positives/negatives possible).

• Theoretically, the power of this test depends on the 8 identifiable probabilities $\\{P(D(0)=d_0, Y(0)=y_0 \mid X=x), P(D(1)=d_1, Y(1)=y_1 \mid X=x) \text{ for }d_0, y_0, d_1, y_1=0, 1\\}.$ Recap that $P(D(0)=a, D(1)=b, Y(0)=c, Y(1)=d \mid X=x):=w_{a,b,c,d}(x)$, by setting $w_{0100}(x), w_{1000}(x), w_{0111}(x), w_{1011}(x)=0$, let $\mathcal U=\\{(w_{0000}(x), w_{1100}(x), w_{0011}(x), w_{1111}(x), w_{ab10}(x), w_{ab01}(x)\text{ for }a, b=0, 1)\mid w_{0000}(x)+w_{1100}(x)+w_{0011}(x)+w_{1111}(x)+\sum_{a, b}w_{ab10}(x)+\sum_{a, b}w_{ab01}(x)=1\\}$ be the unit polyhedron in the 12-dim space with total edge length 1, and denote the linear transformation in Sec. 4.1 from the 12-dim non-zero $w_{a,b,c,d}(x)$ to the above 8-dim identifiable probabilities as $B$. Then the power of this test is $\int_{B(\mathcal U)} dP$.
• Empirically, we add experiments reporting the sensitivity and specificity of our PCF test using various estimators.

The above experimental results align with our theoretical claims for our proposed PCF test in Sec. 4.1, i.e., our PCF test is necessary so that the false positive rate is 0. We also empirically show that our PCF test has a relatively low false negative rate.

More discussion on the two important references

We appreciate your insightful comments!

• Compared with Imai & Jiang (2023), we highlight the following differences.

  • For estimands, they focuses on $(Y(D=0), Y(D=1))$, while we focused on $P(D(A=0), D(A=1)\mid Y(A=0), Y(A=1))$, resulting in different assumptions and techniques.
  • For assumptions, they further assume monotonicity holds, i.e., $Y(D=1)\leq Y(D=0)$, while we only assumes ignorability, which can be relaxed by leveraging the sensitivity analysis such as (Fawkes, Jake, et al., NeurIPS 24).
  • For techniques, their framework is more like instrumental variable or mediation analysis (see Fig. 1 in Imai & Jiang), while we take advantage from linear programming used by (A. Li and J. Pearl, AAAI 22 & 24).

• For Kilbertus et al. (2020), we remark that

  • Principle stratification can be regarded as an unmeasured confounder (see J. Pearl's “Principal Stratification - A Goal or a Tool?” for details), and Kilbertus et al. (2020) studies how fairness constraints are affected by unmeasured confounders;
  • Our paper proposes PCF built on this unmeasured confounder, thus we can compare the difference in performance between PCF and original CF benefiting from Kilbertus et al. (2020).

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Please let us know if you have further questions -- thank you so much!

Thank you for your valuable feedback and the time dedicated to reviewing our work. We address your concerns and questions as follows.

Comparison between CF and PCF

• We can define PCF within the SCM framework, which is equivalent to the potential outcome framework used in our original manuscript.
• For notations, $A$, $X$, and $Y$ are defined the same as in CF [1], and $D$ denotes the decision-making from $A$, $X$, and $Y$, which broadens the $\hat Y$ in CF [1]. In SCM, $Y=f_Y(A, X, \epsilon_Y)$ and $D=f_D(A, X, Y, \epsilon_D)$, so as $Y$ and $\hat Y$ in CF [1].
• For fairness metrics, PCF requires $P(D_{A←a}(U)=y\mid X=x, A=a)=P(D_{A←a'}(U)=y\mid X=x, A=a)$ only for individuals with $Y_{A←a}(U)=Y_{A←a'}(U)$, i.e., $A$ has no effect on $Y$, whereas CF requires to be satisfied for all individuals.
• The main challenges are how to identify individuals with $Y_{A←a}(U)=Y_{A←a'}(U)$ (see Judea Pearl’s "Principal Stratification – A Goal or a Tool?" for more details within SCM, especially Fig. 1 and Table 1), and how to enforce CF on these individuals, instead of all individuals.

More experiments on common datasets

• Motivated by the reviewer, we add experiments comparing more CF methods on the suggested Law and UCI Adult datasets.

• Key observation 1: Our method improves PCF more significantly compared to the original CF metric, in which PCF only focuses $Y(0)≠Y(1)$ but CF focuses on all individuals.

• Key observation 2: Existing CF methods wouldn’t perform better than the proposed approach on the PCF metric.

Causal discovery in experiment

• Our method for imposing PCF does not require causal discovery. Considering there is no DAG in real-word datasets, we implement causal discovery to first obtain CPDAG, then sample a DAG as the ground truth for simulating the counterfactuals.

Necessary condition for testing PCF

• Thank you for your insightful question! Below we supplement both theory and experiments to analyze the power of our proposed test—what is the likelihood that an algorithm violating PCF can pass this test (also known as sensitivity)?
• Theoretically, the power of this test depends on the 8 identifiable probabilities $\\{P(D(0)=d_0, Y(0)=y_0 \mid X=x), P(D(1)=d_1, Y(1)=y_1 \mid X=x) \text{ for }d_0, y_0, d_1, y_1=0, 1\\}.$ Recap that $P(D(0)=a, D(1)=b, Y(0)=c, Y(1)=d \mid X=x):=w_{a,b,c,d}(x)$, by setting $w_{0100}(x), w_{1000}(x), w_{0111}(x), w_{1011}(x)=0$, let $\mathcal U=\\{(w_{0000}(x), w_{1100}(x), w_{0011}(x), w_{1111}(x), w_{ab10}(x), w_{ab01}(x)\text{ for }a, b=0, 1)\mid w_{0000}(x)+w_{1100}(x)+w_{0011}(x)+w_{1111}(x)+\sum_{a, b}w_{ab10}(x)+\sum_{a, b}w_{ab01}(x)=1\\}$ be the unit polyhedron in the 12-dim space with total edge length 1, and denote the linear transformation in Sec. 4.1 from the 12-dim non-zero $w_{a,b,c,d}(x)$ to the above 8-dim identifiable probabilities as $B$. Then the power of this test is $\int_{B(\mathcal U)} dP$.
• In fact, it be proved that such an optimization-based approach can obtain the tightest upper and lower bounds of $P(D(0), D(1), Y(0), Y(1) \mid X=x)$, indicating the optimality of this test.
• Empirically, we add experiments reporting the sensitivity and specificity of our PCF test using various estimators.

Causal discovery might be helpful for PCF

• PCF enforce CF, it would be also interesting to enforce path-specific CF (Chiappa, 2019) on some individuals rather than all. To achieve this, the learned causal diagram via causal discovery can help to estimate the principal strata direct effect (PSDE) to identify these individuals (see also Sec. 4.1 of Pearl’s "Principal Stratification – A Goal or a Tool?").

Interpretation of Theorem 2

• Theorem 2 shows the proposed DR estimator can unbiasedly estimate $P(D(a)=d, Y(a)=y \mid X=x)$ in Sec. 4.1 with large samples.

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We are eager to hear your feedback. We’d deeply appreciate it if you could let us know whether your concerns have been addressed -- thank you so much!