Counterfactual Fairness for Predictions using Generative Adversarial Networks

Responses to "Questions":

1. Correctness of counterfactual mediators

Thank you for raising this important question. You are right in pointing out that, during training, we cannot directly learn the counterfactual mediators in a supervised way as they are unobservable. Instead, we can only leverage the reconstruction loss on the factual mediators. The reason why we can still learn the correct counterfactual mediators is due to the adversarial training process of the generator. By training the discriminator to differentiate between factual and generated counterfactual mediators, the generator is guided to learn the correct counterfactual distribution.

Motivated by your question, we now added a new Lemma 2 where we show theoretically that we learn the correct counterfactual mediators. Specifically, our new Lemma 2 offers a theoretical guarantee that our generator consistently estimates the counterfactual distribution of the mediator. This first provides a theoretical justification behind the design of our generator for the counterfactual mediators. On top of that, Lemma 2 addresses your question by establishing convergence to the correct counterfactual distribution (point mass distribution).

2. Generalization to multiple social groups

Thank you. Our method can easily extended to scenarios with multiple social groups. To demonstrate this, we improved our paper in the following ways:

• We added a description of how our method can be applied to a sensitive attribute with multiple categories (see our new Appendix G). Importantly, we do not need several GANs for each category $a$. Instead, we show how a single GAN is sufficient. The reason for that is that we can learn the joint distribution for all counterfactuals.

• We added mathematical proof of how a single GAN learns the joint distribution for all counterfactuals (see our new Corollary 1).

• We added new experiments to show that our method is also effective in settings with multiple categories (see our new Appendix G4).

We also updated the codes in our anonymous GitHub repository accordingly.

Responses to "Weaknesses"

3. Methods inferring latent variables may hurt prediction performance by introducing bias

Thank you for giving us the opportunity to clarify the benefits of our methods over the baselines. Below, we first explain why even the baseline methods with latent variables are fairly tricky, and we then discuss why our GAN-based approach is superior.

Why is the use of baseline methods based on latent variables tricky?

Importantly, even the baselines for counterfactual fairness are far from being easy as they do not rely on off-the-shelf methods. Rather, they also learn a latent variable in non-trivial ways. More specifically, the inferred latent variable $U$ should be independent of sensitive attribute $A$ while representing all other useful information from the observation data. However, there are two main challenges: (1) The latent variable $U$ is not identifiable. (2) It is very hard to learn such $U$ to satisfy the above independence requirement, especially for high-dimensional or other more complicated settings. Hence, we argue that baselines based on some custom latent variables are highly challenging.

Because of (1) and (2), there are no theoretical guarantees for the VAE-based methods. Hence, it is mathematically unclear whether they actually learn the correct counterfactual fair predictions. In fact, there is even rich empirical evidence that VAE-based methods are often suboptimal. VAE-based methods use the estimated variable $U$ in the first step to learn the counterfactual outcome $\mathbb{P}\left(\hat{Y}_{ a'}(\mathbf{U}) \mid X=x, A=a, M=m \right).$ The inferred, non-identifiable latent variable can be correlated with the sensitive attribute which may harm fairness, or it might not fully represent the rest of the information from data and harm prediction performance.

How our method overcomes the above challenges: We address the above challenges by learning the counterfactuals directly. Thus, our method eliminates the need for a two-step process that learns $U$. To this end, we avoid the complexities and potential inaccuracies of inferring and then using latent variables. More formally, we generate counterfactual samples from the learned counterfactual distribution. For each specific input data $(X=x, A=a,M=m)$, we then generate the counterfactual mediator from the distribution $\mathbb{P}(M_{a'} \mid X=x, A=a, M=m)$. This results in overall more accurate and robust predictions.

More importantly, our new Lemma 2 even provides theoretical guarantees that we learn the correct counterfactual mediators and, therefore, ensure counterfactual fair predictions. To the best of our knowledge, none of the baselines based on latent variables have such guarantees.

As for GAN training, we followed best practices to ensure stability. For example, we employed common techniques for GAN training such as batch normalization, label smoothing, etc. On top of that, counterfactual mediators are commonly low-dimensional in practice. Recall that only a subset of all variables are outputs of our GAN (i.e., the mediators but not the covariates, not the treatment, and not the target variable). This is a crucial difference from many other applications of GANs where the outputs are high-dimensional data such as images and videos. This is one reason why we find our GAN framework to perform well throughout all of our experiments.

We greatly appreciate your insightful feedback! We are thankful that you find our method important and novel.

Responses to "Weaknesses"

1. Applicability beyond binary sensitive attributes

Thank you. Our method can easily extended to scenarios where the sensitive attribute has multiple categories. (We only opted for the binary case for ease of readability.) To this end, we improved our paper in the following ways:

• We added a new description of how our method can be applied to a sensitive attribute with multiple categories (see our new Appendix G). Importantly, we do not need several GANs for each category $a$. Instead, we show how a single GAN is sufficient. The reason for that is that we can learn the joint distribution for all counterfactuals.
• We added a mathematical proof that a single GAN can effectively learn the joint distribution for all counterfactuals (see our new Corollary 1).
• We added new experiments to show that our method is also effective in settings with multiple categories (see our new Appendix G.4).

2. New theoretical guarantees beyond Lemma 1

Thank you. We have greatly reworked our theoretical analysis:

• We added a new Lemma 2 with theoretical guarantees to show that our generator provides consistent estimates of the counterfactual distribution of mediators. This ensures that we learn the correct counterfactuals. Together with Lemma 1, this ensures mathematically that we achieve counterfactual fairness.

• Importantly, the proof of our new Lemma 2 is non-trivial. We kindly refer to our extensive proof in Supplement D.2, where we leverage ideas from [1,2] that were published only recently at NeurIPS 2023.

• The hyperparameter $\lambda$ controls the strength of counterfactual fairness. Hence, by $\lambda$->$\infty$, we ensure that the predictions adhere to counterfactual fairness more strictly. One may think that one could make the hyperparameter $\lambda$ similarly arbitrarily large to ensure counterfactual fairness. However, this would undermine the accuracy of the predictions. Rather, $\lambda$ is a hyperparameter that is used to trade-off counterfactual fairness vs. accuracy.

• To the best of our knowledge, ours is the first neural method that offers theoretical guarantees that the method is effective in learning counterfactual fairness. In fact, many baselines (e.g., [CFGAN]) have even been shown to learn incorrect objectives that act as heuristics but fail to recover counterfactual fairness. To the best of our knowledge, our method is the first neural approach that ensures to correctly learn counterfactual fairness predictions with theoretical guarantees.

References:

[1] Arash Nasr-Esfahany, Mohammad Alizadeh, and Devavrat Shah. Counterfactual identifiability of bijective causal models. In ICML, 2023

[2] Valentyn Melnychuk, Dennis Frauen, and Stefan Feuerriegel. Partial counterfactual identification of continuous outcomes with a curvature sensitivity model. In NeurIPS, 2023.

Responses to "Positioning w.r.t. state of the Art":

3. Addressing concerns about the CFGAN [1]:

Thank you. In response to the query regarding CFGAN and its relation to interventional versus counterfactual fairness, we would like to direct the reviewer's attention to the paper [6]. Therein, the authors show that CFGAN actually satisfies the definition of Discrimination avoiding through causal reasoning. This means: a generator is said to be fair if the following equation holds $P(Y=y \mid X=x, d o(A=a))=P\left(Y=y \mid X=x, d o\left(A=a^{\prime}\right)\right)$. This is different from counterfactual fairness. In counterfactual fairness, a generator producing samples $(X, A, Y)$ with distribution $P$ is said to be counterfactually fair if: $P\left(Y_{a}=y \mid X=x, A=a\right)=P\left(Y_{a’ }=y \mid X=x, A=a\right),$ for all $y \in \mathcal{Y}, x \in \mathcal{X}, a, a’ \in \mathcal{A}$. As such, CFGAN actually considers interventional queries (=level 2 in Pearl’s causality ladder), while counterfactual fairness involves counterfactual queries (=level 3 in Pearl’s causality ladder).

For details of the above, we kindly refer to paper [6], which provides a comprehensive explanation of the actual distribution CFGAN learns and the distinction between counterfactual fairness, along with detailed proofs.

Reassuringly, we remind that CFGAN is vastly different from our method (see our discussion in Appendix B.2). (1) Different tasks: CFGAN is designed for fair data generation tasks, while our model is designed for learning predictors to be counterfactual fairness. Hence, both address different tasks. (2) Different architectures: CFGAN employs two generators, while our method has a single generator and discriminator. Further, fairness enters both architectures at different places. In CFGAN, fairness is ensured through the GAN setup, whereas our method ensures fairness in a second step through our counterfactual mediator regularization. (3) Different training objectives: The training objectives are different: CFGAN learns to mimic factual data. In our method, the generator learns the counterfactual distribution of the mediator through the discriminator distinguishing factual from counterfactual mediators. (4) No theoretical guarantee for CFGAN: Unlike our method, CFGAN does not have theoretical guarantees to correctly learn counterfactual fair predictions. Only our method offers theoretical guarantees for the task at hand. In sum, even though CFGAN also employs GANs, it is vastly different from our method.

Reference:

[6] Mahed Abroshan, Mohammad Mahdi Khalili, and Andrew Elliott. Counterfactual fairness in synthetic data generation. In Neurips 2022 SyntheticData4ML Research, 2022.

Responses to "Positioning w.r.t. state of the Art":

• Questions related to the baselines

2. Addressing concerns about the VAE-based methods:

Thank you. Although a standard Gaussian is a common choice for the prior in VAEs, recent research, such as [2,3], has highlighted that this choice can be suboptimal. The standard Gaussian prior can lead to over-regularization, contributing to poorer density estimation performance. This issue is often referred to as the posterior-collapse phenomenon, as discussed by [4].

Regarding GANs, it is true that they often utilize sampling from a standard Gaussian in the latent space. However, this is not a strict requirement. Various studies have demonstrated effective GAN models that operate without sampling from a Gaussian distribution [5]. Our method diverges from the standard practice as well; in our GAN model, we do not rely on sampling from a standard Gaussian. Instead, we focus on generating counterfactuals based on the learned counterfactual distribution. Thus, our method is more robust by directly learning the transformation.

Motivated by your question, we added a new Lemma 2, which provides theoretical justification for our method. Therein, we prove that our method allows for consistent estimation of the counterfactual distribution. This is important for two reasons. First, it provides a theoretical justification behind the design of our GAN method. Second, it ensures that we correctly learn counterfactual fairness. To the best of our knowledge, ours is the first neural method for counterfactual fairness that offers such guarantees. Importantly, this is a major difference to the VAE-based baselines, which do not have similar guarantees.

References:

[1] Depeng Xu, Yongkai Wu, Shuhan Yuan, Lu Zhang, and Xintao Wu. Achieving causal fairness through generative adversarial networks. In IJCAI, 2019.

[2] Takahashi, Hiroshi and Iwata, Tomoharu and Yamanaka, Yuki and Yamada, Masanori and Yagi, Satoshi. "Variational autoencoder with implicit optimal priors." In AAAI, 2019.

[3] Hoffman, Matthew D., and Johnson, Matthew J. ELBO surgery: yet another way to carve up the variational evidence lower bound. In Workshop in Advances in Approximate Bayesian Inference, NeurIPS. 2016.

[4] Aaron van den Oord, Oriol Vinyals, koray kavukcuoglu. Neural Discrete Representation Learning. In In NeurIPS.2017

[5] Ledig C, Theis L, Huszár F, et al. Photo-realistic single image super-resolution using a generative adversarial network. Proceedings of the IEEE conference on computer vision and pattern recognition. 2017

We appreciate the positive feedback on our paper as well as the suggestions for further improvement! As you will see below, we took your feedback at heart and have carefully and thoroughly revised our paper to incorporate your comments.

In addition to comments, we have also added a new theoretical analysis. Specifically, we added a new Lemma 2 to show that our method correctly identifies the counterfactual mediators. Specifically, we prove that our method allows for consistent estimation of the counterfactual distribution. The lemma thus gives a guarantee on the correctness of the generated counterfactual mediators and, therefore, also a guarantee on the upper bound in Lemma 1.

Our new theoretical analysis is important for two reasons. First, it provides a theoretical justification behind the design of our method. Second, it ensures that we correctly learn counterfactual fairness. To the best of our knowledge, ours is the first neural method that offers such guarantees. Note that the VAE-based baselines do not have similar theoretical support, which provides an additional theoretical explanation why our method is effective and even superior.

Responses to "Positioning w.r.t. state of the Art":

• Questions related to the baselines

1. Addressing concerns about the VAE-based methods

Thank you. We followed your suggestion and added the reference to VAE-based methods with regularization (e.g. Grari et al.2023) that seek to remove correlations. However, we would still clarify why VAE-based methods for our task can be problematic and why our proposed method is superior – both theoretically and numerically.

*Why are VAE-based methods not optimal? *VAE-based methods learn a latent variable in non-trivial ways. More specifically, the inferred latent variable $U$ should be independent of sensitive attribute $A$ while representing all other useful information from the observation data. However, there are two main challenges: (1) The latent variable $U$ is not identifiable. (2) it is very hard to learn such $U$ to satisfy the above independence requirement, especially for high-dimensional or other more complicated settings. As a result, VAE-based methods act as heuristics, and, hence, there is no theoretical guarantee that VAE-based methods are actually effective. That means, there is no theoretical guarantee on the fairness level in the final prediction.

Because of (1) and (2), there are no theoretical guarantees for the VAE-based methods. Hence, it is mathematically unclear whether they actually learn the correct counterfactual fair predictions. In fact, there is even rich empirical evidence that VAE-based methods are often suboptimal. VAE-based methods use the estimated variable $U$ in the first step to learn the counterfactual outcome $\mathbb{P}\left(\hat{Y}_{ a'}(\mathbf{U}) \mid X=x, A=a, M=m \right).$ The inferred, non-identifiable latent variable can be correlated with the sensitive attribute which may harm fairness, or it might not fully represent the rest of the information from data and harm prediction performance.

How our method overcomes the above challenges: We address the above challenges by learning the counterfactuals directly. Thus, our method eliminates the need for a two-step process that learns $U$ but, instead, we learn $U$ directly in a single step through our GAN. To this end, we avoid the complexities and potential inaccuracies of inferring and then using latent variables. More formally, we generate counterfactual samples from the learned counterfactual distribution. For each specific input data $(X=x, A=a,M=m)$, we then generate the counterfactual mediator from the distribution $\mathbb{P}(M_{a'} \mid X=x, A=a, M=m)$. This results in overall more accurate and robust predictions. Further, our new Lemma 2 even shows that we correctly learn the counterfactual mediators and, unlike VAE-based methods, we thereby prove that our method is effective in achieving counterfactual fairness.

Responses “Soundness of the objectives”

1.Addressing concern about the spurious effect

Thank you for asking questions about the potential spurious effects in the correlation between $X$ and $A$, which can have an impact on $Y$ We welcome the opportunity to provide a clarification. It is true that if $X$ and $A$ are correlated, there would occur a spurious effect ($A$->$X$->$Y$) in our causal graph. However, note that the counterfactual fairness notion [8] only addresses direct effect ($A$->$Y$) and indirect effect ($A$->$M$->$Y$), but does not include the spurious effect ($A$->$X$->$Y$). This is because the spurious effect ($A$->$X$->$Y$) does not influence the counterfactual fairness in the prediction.

We kindly refer to paper [9], which provides an in-depth and detailed discussion of the difference between various fairness notions and causal effects. Figure 4.5 in paper [9] explains clearly that counterfactual fairness is the unit-level fairness notion and, thus, does not include spurious effects. This makes counterfactual fairness different from other fairness notions (such as causal fairness). To illustrate the above, let us draw upon your example. In the given scenario, $X$ represents age, and $A$ represents gender. We further assume that there is correlation between them ( typically, women are older than men). A company aiming for gender fairness, thus chooses not to use gender directly in its hiring decisions. Since age is not a sensitive attribute, they use age for making decisions, preferring younger candidates. This method, although seemingly unbiased w.r.t gender, inadvertently leads to a higher number of male employees than female employees due to the correlation between age and gender. This example points to the reviewer's concern: such correlations do indeed influence outcomes at the population level, creating an average effect.

However, from the perspective of counterfactual fairness, the focus is on the individual level. For each person, it examines the situation where altering an individual's gender (e.g., from male to female) wouldn't change the employment outcome of this person. Thus, even though a correlation between age and gender exists, it does not affect each individual's final employment decision. Therefore, having correlations between $X$ and $A$ does not affect the counterfactual level in our prediction in Y, which is consistent with our setting.

2.Addressing concerns about the adversarial loss

Thank you. We would like to clarify that our adversarial objective is indeed symmetric, addressing counterfactual scenarios for both $A=1$ and $A=0$ classes. There is no preference for the class and the loss formula is correct. Since the sum of the discriminator output is 1 (i.e., a sum of the probability), we merge the two items in the original version of the loss equation into one (see our updated Eq. 6 for the loss). If you view $A$ as the true label, it is exactly how cross-entropy loss is computed. The counterfactual output of the generator is always fed into the discriminator. The generator of the GAN then minimizes the conditional propensity-weighted Jensen–Shannon divergence (JSD). To show the above more explicitly, we also added a new Lemma 2 and a corresponding proof. We kindly refer the reviewer to our proof, which should make the above more clear. (see Eq. (31) in Appendix D.2).

References:

[9] Drago Plecko and Elias Bareinboim. Causal fairness analysis. In arXiv preprint, 2022.

Responses to “experiments”

Thank you. We followed your suggestions and included several new experiments (see new Appendix X.1 and X.2).

1. (a). Correlation Between $X$ and $A$:

We added a new experiment where we included correlation between $X$ and $A$. The experiment results show again that we achieved the best performance (see our new Appendix H.1). Importantly, these results confirm that introducing a correlation between $X$ and $A$ does not adversely affect the counterfactual level of our predictions for $Y$. This finding aligns with our above response to the initial concern on spurious effects, showing that having correlation between X and A does not affect the counterfactual fairness level in our prediction in Y.

(b). Generation of M from X and A:

Thank you for asking the questions. We would like to clarify that $M$ is not deterministically deduced from $X$ and $A$. In our dataset, the generation of $M$ incorporates stochastic elements, specifically through the inclusion of noise $U_M$, which makes the process non-deterministic. We have thus revised our paper to spell out explicitly that the generation of $M$ is non-deterministically. In response to your query regarding the evaluation of counterfactual fairness: This limitation is widely recognized in the field, and as a result, the use of synthetic or semi-synthetic datasets for such evaluations is a well-established norm, as is the quasi-standard in the literature [7,8,10].

Regarding the simplicity of the setting: our setting is intentionally designed analogous to prior research [8, 10] and especially [7]. Therein, the setting for modeling fairness violations is similar to ours, so that we allow for better comparability. Lastly, it is crucial to note that our prediction model does not explicitly utilize the sensitive attribute. The inputs for our predictor $h$ are $X$ and $M$. This is done to adhere to ethical guidelines in practice and to reduce the risk of biased outcomes. We have revised our paper to spell out clearly that we follow your comment and do not use the sensitive attribute in our predictor.

2. We followed your suggestion and added new performance comparisons for our real-world dataset (see our new Appendix H.2). Due to the unavailability of counterfactuals for real-world data, we use our generated counterfactual as the ground-truth counterfactual and show the utility function value as on synthetic datasets. As a result, we find that our proposed method is highly effective.

3. Thank you. We paid great attention to make our work self-contained and reproducible. Wherever possible, we used the original source codes from the authors for fair comparison (and we contacted all authors whenever the code was not public in the first place). To address your comment, we have improved our papers as follows: We extended our implementation details and now state clearly where the baseline codes are from (see our revised Appendix F.1). We have also added detailed descriptions including pseudocode for ADVAE and CFGAN (see our new materials in Appendix I). We have made both our code and all datasets publicly available (see the GitHub link in our paper). Thereby, we ensure that all experiments reported in our study are indeed self-contained and reproducible.

Reference:

[7] Ioana Bica, James Jordon, and Mihaela van der Schaar. Estimating the effects of continuous-valued interventions using generative adversarial networks. In NeurIPS, 2020.

[8] Matt J Kusner, Joshua Loftus, Chris Russell, and Ricardo Silva. Counterfactual fairness. In NeurIPS, 2017.

[10] Francesco Quinzan, Cecilia Casolo, Krikamol Muandet, Niki Kilbertus, and Yucen Luo. Learning counterfactually invariant predictors. In arXiv preprint, 2022.

Thanks for this reference but I stil think that CFGAN proposes a way to achieve counterfactual fairness, not only interventional one, as they produce both versions of the individuals, with modified Y and X according to the sensitive attribute (using same Z as seed of the generation). From this it is possible to learn a predictor that achieves comparable outcomes for both versions...

Addressing concerns about the adversarial loss

Thank you for your question. We would like to clarify that the updated version of adversarial loss in Eq.6 is *exactly the same as the original version (but with better notation). We would also like to explain below why our formalization of the loss is correct.

In the original version, our adversarial loss is written as ${L}\_{adv}=\mathbb{E}\_{({X,A,M)\sim \mathbb{P}\_\mathrm{f}}}\left[A \log\big( {D}(X, \tilde{G}\left(X,A,M \right))\_{A}\big)+A^{\prime} \log \big( 1-{D}(X, \tilde{G}\left(X,A,M \right))_{A'}\big)\right]$.

Our current loss is ${L}\_{adv} = \mathbb{E}\_{({X,A,M)\sim \mathbb{P}\_\mathrm{f}}}\left[\log\big( {D}(X, \tilde{G}\left(X,A,M \right))_{A}\big) \right ]$.

It might not look that straightforward to see the updated version is exactly the same (just merge the original two terms into one), but it is actually the same as the previous one. Let us clarify that step by step.

In binary setting, $1-{D}(X, \tilde{G}\left(X,A,M \right))_{A'}$ is actually equal to ${D}(X, \tilde{G}\left(X,A,M \right))\_{A}$ . This because the sum of ${D}(X, \tilde{G}\left(X,A,M \right))\_{A’}$ and ${D}(X, \tilde{G}\left(X,A,M \right))\_{A}$ should be 1 (i.e., he discriminator $D$ outputs the probability and the sum of the total probability is 1).

Let’s consider $A=1$ and $A=0$ respectively, for the original version of loss.

• When $A=1$, $A’ = 0$ makes the second term vanish, then the only thing left is $\log({D}(X, \tilde{G}\left(X,A,M \right))\_{A})$, which is $\log({D}(X, \tilde{G}\left(X,1,M \right))\_{1})$.

• When $A=0$, then the first term turns to 0 and the second term is $\log(1-{D}(X, \tilde{G}\left(X,A,M \right))_{A'})$, which is equal to $\log({D}(X, \tilde{G}\left(X,A,M \right))\_{A})$ , and the loss now is $\log({D}(X, \tilde{G}\left(X,0,M \right))\_{0})$ .

Our current loss is $\mathbb{E}\_{({X,A,M)\sim \mathbb{P}\_\mathrm{f}}}\left[\log\big( {D}(X, \tilde{G}\left(X,A,M \right))_{A}\big) \right ]$. Let’s consider $A=1$ and $A=0$ respectively, for the current version of loss.

• When$A=1$, the loss is $\log({D}(X, \tilde{G}\left(X,1,M \right))\_{1})$

• When $A=0$, the loss is $\log({D}(X, \tilde{G}\left(X,0,M \right))\_{0})$.

So our updated version is exactly the same as the original version.

We want to clarify it considers the counterfactual in our formula. The output of the generator $\tilde{G}\left(X,A,M \right)$ is always fed into the discriminator. Besides, we consider both classes for $A=1$ and $A=0$. The generator of the GAN actually minimizes the conditional propensity-weighted Jensen–Shannon divergence (JSD). To show the above more explicitly, we kindly refer the reviewer to our proof, which should make the above more clear. (see Eq. (31) in Appendix D.2).

For ease of understanding, one can simply view $A$ as the true label in a classification class, and our loss is no different from the normal cross-entropy loss for the classification task.

Action (in blue font): We added a brief intuition along the above lines to explain our loss in greater detail.

We are again thankful for your question. We hope that our answer helped in addressing your questions satisfactorily. Please let us know if there are further things in our presentation that should be improved.

Addressing concern on “identifiable for VAE-based method”:

We break down the reviewer's questions and answer it point by point below:

1. What does identification mean:

First, “identification” does not mean “estimation“. In causal inference, you can think about identifiability as the mathematical condition that permits a causal quantity to be measured from observed data. We kindly refer to papers [2][3] which all discuss clearly what identification means in causality. Importantly, identification is different from estimaton because methods that act as heuristics may return estimates but they do not correspond to the true value. (see paper [1] where the authors provide several concerns that, if it is not unique, it is possible to have local minima, which leads to unsafe results.

2. Non-identifiable for VAE based methods have been shown in prior works of literature:

We kindly refer the reviewer to this paper [7] [Neural Causal Models for Counterfactual Identification and Estimation] ” In this paper, the authors show that VAE-based counterfactual inference do not allow for identifiability. The results directly apply to variational inference-based methods they listed (VACA Sanchez-Martin et al. (2021), DeepSCM Pawlowski et al. (2020)), which do not have proper identification guarantees. Also, the result from non-linear ICA (which is the task of variational autoencoders) shows that the latent variables are non-identifiable [6]. In simple words, VAE-based methods can estimate the latent variable but it is not guaranteed that it can be correctly identified, leading to risks that the latent variable is often estimated incorrectly.

3. Non-identifiability of the latent variables means non-identifiability of the counterfactual queries:

We kindly refer to paper [4] (Partial Counterfactual Identification of Continuous Outcomes with a Curvature Sensitivity Model. ) which says that the non-identifiability of the latent variables means non-identifiability of the counterfactual queries. Hence, VAE-based methods can not ensure that they correctly learn counterfactual fairness, only our method does so.

4. Regarding our method

We diverge from the above practices. Instead, we focus on generating counterfactuals based on the learned counterfactual distribution. Thus, our method is more robust by directly learning the transformation. More importantly, we provide theoretical guarantees in our paper that we correctly learn counterfactual fairness (see Lemma 1 and Lemma2)

• How our method overcomes the above challenges:

We address the above challenges by learning the counterfactuals directly. Thus, our method eliminates the need for a two-step process that learns $U$. To this end, we avoid the complexities and potential inaccuracies of inferring and then using latent variables. More formally, we generate counterfactual samples from the learned counterfactual distribution. For each specific input data $(X=x, A=a,M=m)$, we then generate the counterfactual mediator from the distribution $\mathbb{P}(M_{a'} \mid X=x, A=a, M=m)$. This results in overall more accurate and robust predictions. Further, our new Lemma 2 even shows that we correctly learn the counterfactual mediators and, unlike VAE-based methods, we thereby prove that our method is effective in achieving counterfactual fairness.

Action (in blue font): We added a clarification to our paper to distinguish identifiability vs. estimatability (see our revised Section 2 and Appendix B.1). We further spell out clearly that one weakness of the VAE-based methods is that they are non-identifiability, while our method allows for identifiability.

We are again thankful for your question. We hope that our answer helped in addressing your questions satisfactorily. Please let us know if there are further things in our presentation that should be improved.

Reference:

[1] D’Amour A. On Multi-Cause Causal Inference with Unobserved Confounding: Counterexamples[J]. Impossibility, and Alternatives, 2019.

[2] Pearl, Judea. Causal inference in statistics: An overview. 2009

[3]Peters, Jonas, Dominik Janzing, and Bernhard Schölkopf. Elements of causal inference: foundations and learning algorithms. The MIT Press, 2017.

[4]Melnychuk, Valentyn, Dennis Frauen, and Stefan Feuerriegel. Partial Counterfactual Identification of Continuous Outcomes with a Curvature Sensitivity Model. Neurips 2023.

[5]De Brouwer, Edward. "Deep Counterfactual Estimation with Categorical Background Variables." Neurips. 2022.

[6]Khemakhem I, Kingma D, Monti R, et al. Variational autoencoders and nonlinear ica: A unifying framework. International Conference on Artificial Intelligence and Statistics. 2020.

[7]Xia, Kevin, Yushu Pan, and Elias Bareinboim. "Neural causal models for counterfactual identification and estimation." arXiv 2022.

Responses to "Weaknesses"

• Response to: Differences from CFGAN [1]

We appreciate the opportunity to clarify that our method is significantly different from CFGAN (in terms of task, architecture, learning objectives, etc.). We highlight these differences below:

Different tasks:

CFGAN is designed for fair data generation tasks, while our model is designed for learning predictors to be counterfactual fairness. Hence, both address different tasks.

Different architectures:

CFGAN employs two generators, each aimed at simulating the original causal model and the interventional model, and two discriminators, which ensure data utility and causal fairness. We only employ a streamlined architecture with a single generator and discriminator. Further, fairness enters both architectures at different places. In CFGAN, fairness is ensured through the GAN setup, whereas our method ensures fairness in a second step through our counterfactual mediator regularization.

Different training objectives:

The training objectives are different: CFGAN learns to mimic factual data. In our method, the generator learns the counterfactual distribution of the mediator through the discriminator distinguishing factual from counterfactual mediators.

No theoretical guarantee for CFGAN:

CFGAN is proposed to synthesize a dataset that satisfies counterfactual fairness. However, a recent paper [2] has shown that CFGAN is actually considering interventions (=level 2 in Pearl’s causality ladder) and not counterfactuals (=level 3). Hence, CFGAN does not fulfill the counterfactual fairness notion, but a different notion based on do-operator (intervention). For details, we refer to reference [2], Definition 5 therein, called ``Discrimination avoiding through causal reasoning''): A generator is said to be fair if the following equation holds: for any context $A=a$ and $X=x$, for all value of $y$ and $a' \in \mathcal{A}$, $P(Y=y \mid X=x, d o(A=a))=P\left(Y=y \mid X=x, d o\left(A=a' \right)\right)$, which is different from the counterfactual fairness $P(Y_a=y \mid X=x, A=a)=P\left(Y_{a'}=y \mid X=x, A=a\right)$. (We also discuss CFGAN considers interventions not counterfactual in our paper see Appendix. I.1) Moreover, CFGAN lacks theoretical support for its methodology (no identifiable guarantee or counterfactual fairness level). In contrast, our method strictly satisfies the principles of counterfactual fairness and provides theoretical guarantees on the counterfactual fairness level. In sum, only our method offers theoretical guarantees for the task at hand.

Suboptimal performance of CFGAN:

Even though CFGAN can, in principle, be applied to counterfactual fairness prediction, it is suboptimal. The reason is the following. Unlike CFGAN, which generates complete synthetic data under causal fairness notions, our method only generates counterfactuals of the mediator as an intermediate step, resulting in minimal information loss and better inference performance than CFGAN. Furthermore, since CFGAN needs to train the dual-generator and dual-discriminator together and optimize two adversarial losses, it is more difficult for stable training, and thus its method is less robust than ours.

In sum, even though CFGAN also employs GANs, it is vastly different from our method.

Action: We carefully discuss the differences between CFGAN and our method in our revised paper (see Appendix B.2 and I.1). Therein, we spell out clearly that CFGAN is different in terms of task, architecture, and theoretical guarantees.

References

[1] Depeng Xu, Yongkai Wu, Shuhan Yuan, Lu Zhang, and Xintao Wu. Achieving causal fairness through generative adversarial networks. In IJCAI, 2019.

[2] Mahed Abroshan, Mohammad Mahdi Khalili, and Andrew Elliott. Counterfactual fairness in synthetic data generation. In Neurips 2022 SyntheticData4ML Research, 2022.

Thank you for your helpful review! We appreciate that you find our paper important, rich, and well-organized. We have carefully addressed your comments below to improve our work.

Responses to "Weaknesses"

• Response to: New theoretical guarantees

Thank you for your suggestion. We have entirely reworked our theoretical analyses:

We now added a new theoretical analysis to show our method’s effectiveness (see our new Lemma 2). Our new Lemma 2 states that our method correctly identifies the counterfactual mediators. Specifically, we prove that our method allows for consistent estimation of the counterfactual distribution, $\mathbb{P}(M_{a'} \mid X=x, A=a, M=m)$. Lemma 2 thus gives a guarantee on the correctness of the generated counterfactual mediators and, therefore, also a guarantee on the upper bound in Lemma 1, which thus confirms the effectiveness of our method.

We added a new Corollary 1 to show that the above theoretical results also generalize to high-dimensional settings. Our new theoretical analysis is important for two reasons. First, it provides a theoretical justification behind the design of our method. Second, it ensures that we correctly learn counterfactual fairness. To the best of our knowledge, ours is the first neural method for counterfactual fairness that offers such guarantees.

• Response to: Theoretical guarantee to show our method’s effectiveness

Thank you for your helpful comment. We added a new Lemma 2 to give a guarantee on the error of the generated counterfactual ($||M_{A'} - \hat M_{A'}||^2_2$), which is the first term of upper bound terms in Lemma 1. Therefore, when the counterfactual mediator regularization ${R}_\mathrm{{cm}}$ (=the second term of the upper bound), is minimized, the counterfactual fairness level is guaranteed through Lemma1. As such, we achieve a theoretical guarantee that our method is effective in achieving counterfactual fairness. To the best of our knowledge, ours is the first neural method for counterfactual fairness that offers such guarantees.

Thank you for your thorough and insightful feedback! We are glad to hear that you appreciate the strong empirical performance and the clear presentation of our paper.

Response to Weaknesses

Identification of counterfactual mediators

Thank you. We recognize the importance of discussing the identification of counterfactual mediators. To address your point, we have added a new theoretical analysis. Specifically, we have revised our manuscript in the following ways:

• We added a new Lemma 2 to show that our method correctly identifies the counterfactual mediators. Specifically, we prove that our method allows for consistent estimation of the counterfactual distribution. The lemma thus gives a guarantee on the correctness of the generated counterfactual mediators and, therefore, also a guarantee on the upper bound in Lemma 1.

• We added a new Corollary 1 to show that the above theoretical results also generalize to high-dimensional settings. Our new theoretical analysis is important for two reasons. First, it provides a theoretical justification behind the design of our method. Second, it ensures that we correctly learn counterfactual fairness. To the best of our knowledge, ours is the first neural method that offers such guarantees.

Theoretical Guarantee and Counterfactual Estimation

Thank you. We address your comment point-by-point。

Response to: No theoretical guarantee

To address your point, we now provide new theoretical guarantees of the correctness of the generated counterfactual mediators (see our new Lemma2). Our new Lemma 2 shows that our generator consistently estimates the counterfactual distribution of the mediator. It gives a guarantee on the correctness of the generated counterfactual mediators. Together with Lemma 1, it provides theoretical results that our method is effective.

Response to: Correctness of the generated counterfactual mediator

Our new Lemma 2 states that our GAN-based approach consistently estimates the counterfactual distribution of mediators. By converging to the counterfactual distribution (point mass distribution), our generator actually learns a deterministic function that transforms from factual to counterfactual: $\mathbb{P}(M_{a'} \mid X=x, A=a, M=m)$. Under consistency, the generated counterfactual can converge to the ground-truth counterfactual. This confirms the effectiveness and validity of the generated counterfactual mediators.

Response to: Addressing the upper bound in Lemma 1

We acknowledge that Lemma 1 was not sufficient to show the effectiveness of our method. As a remedy, our new Lemma 2 gives a guarantee on the error of the generated counterfactual ($||M_{A'} - \hat M_{A'}||^2_2$), which is the first term of upper bound terms in Lemma 1. Then, by minimizing the loss function ${L}(h)=L_{ce}(h)+\lambda R_{cm}(h)$ , the counterfactual mediator regularization $R_{cm}$ is minimized accordingly, which is the second term in the of upper bound terms in Lemma 1. As a result, we can successfully address your comment: by minimizing the loss, we can guarantee that the counterfactual fairness level is minimized.

Response to Questions

• Fig.1 Interpretation:

The dashed line in Fig.1 illustrates that a correlation between $X$ and $A$ is permissible/allowed in our framework, but a direct causal relationship $X$->$A$ is not a necessity. Note that, if there is no dashed edge between $X$ and $A$, it is actually a stronger assumption, because it forbids the edge between $X$ and $A$ to have any hidden confounders. However, our setting is more general and allows the existence of the confounders. Notwithstanding, we added new experimental results (see new results in Appendix H.1), where we analyze the performance of our method in the presence of confounders, finding that our method is highly effective.

• Choice of mediator variables:

The selection of mediator variables, $M$, typically relies on domain knowledge, following established practices in literature such as [3, 4, 5, 6, 7]. We fixed that sentence in our revised paper.

• Inconsistent notation:

Thank you! The notation $M_{A-\leftarrow a}$ was an error. We have streamlined our notation so that we now consistently use $M_a$ as the potential outcome in the updated manuscript.

• Parenthesis mismatch in Eq. (9):

Thank you. We fixed this.

• Adult dataset:

Thank you for your suggestions. We added a more detailed discussion of our setting to Appendix E.3. Below, we discuss how we selected mediators and the counterfactual fairness metric.

Mediator selection:

We follow common choices in prior research (such as [3, 4, 5, 6]). We consider 'gender' as a sensitive attribute. Mediators are variables that can potentially be influenced by the sensitive attribute. The 'marital status', 'education level', 'occupation', 'hours per week', and 'work class' are all known to be influenced by ‘gender’, and, therefore, we treat them as ‘mediators’. We classified all other variables as covariates.

Counterfactual fairness metric:

We use the Adult dataset to demonstrate the applicability of our method to real-world data, where ground-truth counterfactuals are unavailable. Nevertheless, to understand the implications for counterfactual fairness empirically, we use the generated counterfactuals to measure counterfactual fairness on the test dataset. This allows us to explore the accuracy-fairness trade-off. We have revised our paper to spell out clearly how we computer the counterfactual fairness metric for real-world data. Thereby, we further highlight that our choice is in line with other works from causal fairness as the purpose of the Adult dataset is to demonstrate real-world applicability (and not benchmarking).

Reference

[3] Razieh Nabi and Ilya Shpitser. Fair inference on outcomes. In AAAI, 2018.

[4] Hyemi Kim, Seungjae Shin, JoonHo Jang, Kyungwoo Song, Weonyoung Joo, Wanmo Kang, and Il.Chul Moon. Counterfactual fairness with disentangled causal effect variational autoencoder. In AAAI, 2021.

[5] Depeng Xu, Yongkai Wu, Shuhan Yuan, Lu Zhang, and Xintao Wu. Achieving causal fairness through generative adversarial networks. In IJCAI, 2019.

[6] Francesco Quinzan, Cecilia Casolo, Krikamol Muandet, Niki Kilbertus, and Yucen Luo. Learning counterfactually invariant predictors. arXiv preprint, 2022.

[7] Drago Plecko and Elias Bareinboim. Causal fairness analysis. In arXiv preprint, 2022.

Response to Questions

• Discussion regarding the literature [1] & [2]:

Thank you. We appreciate the opportunity to spell out how our work is different from [1] and [2]. We also have included references to papers [1] & [2] in our revised paper together with a brief discussion. Both [1] & [2] make important contributions to our theoretical understanding of causal fairness. As such, they are orthogonal to our work where we develop a new learning algorithm for counterfactual fair predictions with theoretical guarantees.

Regarding [1]:

The setting of the paper [1] is similar to ours, only some notations are different. In [1], what is referred to as $X(a)$ aligns with $M_a$ in our paper. However, there are several differences in the assumptions that distinguish [1] from our paper.

1. Confounder inclusion:

Our model allows for the existence of the confounders between the sensitive attribute $A$ and its descendant $M$, denoted as $X$ in our paper. This is unlike [1], where such variables are not explicitly considered, so that our setting can be considered to be more general, For example, consider Figure 1 in reference [1]. Therein, the authors aim to handle additional variables (“covariates”) that are not the sensitive attribute but also can affect the mediators like "Age”. The “Age” variable, even when there are no other sensitive attributes such as “race” or “gender” there, still impacts mediators like “GPA”, “LSAT”, and “FYA”. However, such variables are not considered in [1] setting. In contrast, we can put such variables into our covariate $X$. Therefore, we argue that our setting is applicable to a wider range of scenarios.

2. Selection bias: If there exists a collider $S$ between $A$ and $M$ (such that $A \rightarrow M$, $A \rightarrow S$, $M \rightarrow S$), then conditioning on $S$ introduces a correlation between $A$ and $M$. This can be interpreted as the presence of a confounder (such that $A \rightarrow M$, $X \rightarrow A$, $X \rightarrow M$). In [1], the dataset is collected without selection bias, which implies that $A$ and $M$ are unconfounded. This is consistent with our assumptions. In [1], structural counterfactuals satisfy ignorability ($M_a \perp\kern-5pt\perp A$). In our work, since we allow for covariates $X$, the potential outcome $M_a$ is conditionally independent of $A$ given the covariates $X$ ($M_a \perp\kern-5pt\perp A | X$) .

Regarding [2]:

We highly appreciate the theoretical insights from [2], yet there are significant differences in our work. DP is a non-causal fairness notion. Specifically, DP says the target $Y$ should satisfy the conditional distribution $P(Y \mid A=a)=P(Y \mid A=a’)$, and, thereby, it essentially prohibits the use of any variable correlated with the sensitive attribute $A$ for predictions. This includes $A$ itself, its descendants, causes, and any correlated variables. As we discuss below, prohibiting the use of any variable correlated with the sensitive attribute may be fairly restrictive in practice.

In our causal graph, to achieve fairness under DP, one must exclude not only the sensitive attribute $A$ and mediators $M$ but also any variables in $X$ that directly cause or correlate with A. For instance, consider an employment system in a company. If ‘'gender' is the sensitive attribute and 'age' correlates with gender due to regional cultural norms, DP would forbid the use of 'age' for fair decision-making. Our method, however, allows for such usage.

Consequently, achieving DP often results in a significant loss of predictive accuracy. In contrast, our method ensures counterfactual fairness, while not being bound by these strict restrictions, so that our method can achieve a better prediction performance. We performed a few tests using datasets where $X$ and $A$ are uncorrelated. We used the same classifier for better comparability. A method that needs to achieve DP only records an accuracy of 0.76 (i.e., by only using $X$). In contrast, our method achieves an accuracy 0.94 (by allowing the use of $X$ and $M$). In summary, DP has more strict requirements for using variables, thus sacrificing the accuracy of predictions. This is especially relevant when $M$ contains a lot of useful information for prediction. We thus believe that our method is therefore of great value in practice where not only counterfactual fairness but also a good prediction performance is important.

Reference

[1] Fawkes, Jake, Robin Evans, and Dino Sejdinovic. "Selection, ignorability and challenges with causal fairness." In Conference on Causal Learning and Reasoning. PMLR, 2022.

[2] Rosenblatt, Lucas, and R. Teal Witter. "Counterfactual fairness is basically demographic parity." In AAAI. 2023.

Thanks for the revision. I updated my score. But I still have some questions.

1. Can the authors make it clear whether the first statement of Lemma 2 is their contribution or from existing work? With statement 1, the counterfactual distribution is identified. Then it is not surprising we can fit a GAN or other generative models to estimate the counterfactual mediators.

2. Intuitively, I still think the proposed GAN can only generate good factual data as it is never trained on counterfactual data just like the original GAN can only generate good data from the original training distribution. This implies that the strategy of this work is to mak the distribution of counterfactual mediators equivalent to its corresponding conditional distribution with a different value of $A$. With Lemma 2, it basically says, if the conditional $P(M|X=x,A=a')$ can be estimated precisely, then the counterfactual $P(M_{a\leftarrow a'}|X=x,A=a)$ can be estimated accurately because they are the same thing under the assumptions. In this case, my question is, if we can observe enough samples with $(X=x,A=a')$, why is $M_{a\leftarrow a'}|X=x,A=a$ still a counterfactual quantity?