We are grateful for the review! We appreciate the reviewer’s constructive feedback and approved our paper accordingly.

Response to weaknesses

1) Contribution and novelty:

Thank you for giving us the opportunity to emphasize the novelty of our framework. Thereby, we clarify that we do not simply combine two concepts but that our derivations are non-trivial.

Why is the derivation of our bounds non-trivial? We have derived tailored bounds for counterfactual fairness under unobserved confounding. Importantly, our bounds require a novel and careful derivation. In other words, our bounds are not just a simple result of adopting a sensitivity model but require a new derivation tailored to our setting. The reason is the following: Causal fairness notions commonly contain nested counterfactuals and are thus located on level three of Pearl’s ladder of causality. The GMSM (and other sensitivity models) incorporate single-intervention counterfactuals and are thus located on level two. Because of that, existing bounds from sensitivity models are not applicable. In other words, we can not simply adopt existing bounds from sensitivity models, but we need a new derivation for level three of Pearl’s causality ladder.

What is our contribution? We agree that causal fairness and partial identification bounds under the MSM/GMSM have separately been studied before. One of our contributions is the transfer of concepts from sensitivity analysis to the causal fairness literature (note: due to the reasons above, this is non-trivial): As a result, we offer a new framework to assess causal fairness under unobserved confounding. This is a major difference to the existing literature on causal fairness, which has ignored the fairness implications due to unobserved confounding. In other words, we fill an important research gap in the literature (see our new Table 1). Furthermore, we provide important implications for practice: we call for a more cautious use of causal fairness since, due to unobserved confounding, important fairness issues may still be present, because of which fairness in practice can be undermined. As a remedy, we contribute for the first time a framework that can remove potential issues in causal fairness due to unobserved confounding. We thus expect that our findings, as well as our framework, are of immediate practical importance.

How is our framework novel compared to sensitivity analysis? While we adopt a sensitivity analysis as the basis for deriving our bound, we add over sensitivity analysis in several ways. (1) We derive new bounds that are tailored to our bounds. The bounds do not directly follow from the sensitivity analysis (which is unlike other applications of sensitivity models in the literature); instead, a tailored derivation is needed that is non-trivial. (2) We develop an end-to-end framework for fair prediction models. As such, Steps 1 and 2 in our framework are novel and make large contributions to existing sensitivity analysis.

Actions: We improved our work in the following ways to clarify our novelty and why our contributions are non-trivial:

• We included a new Table 1, where we highlight the novelty of our work over existing literature streams. Importantly, ours is the first work to study causal fairness under unobserved confounding.
• We revised the presentation of our framework to explain why existing bounds from sensitivity models are not applicable (as our causal query is in ladder three and not in ladder two of Pearl’s causality ladder). As a result, we state clearly why our bounds are non-trivial (see revised Section 2).
• We added a technical background explaining the theoretical differences between existing bounds from sensitivity models (=ladder two) and our causal query (=ladder three). Thereby, we motivate why a new derivation is needed (see new Supplement C.3).

2) Broader scope across fairness notions:

Thank you for this important suggestion. Our framework is specially designed to incorporate path-specific effects. Nevertheless, we want to emphasize that our framework is general and can also be applied to other fairness notions. To demonstrate that, we added a new Supplement F, where we derive bounds for other notions. Thereby, we demonstrate how our framework can be applied to further fairness notions, which shows the broad applicability of our framework. Thank you as well for pointing out the ambiguity of the name “path-specific causal effects”. We rephrased the notation in our paper and now refer to the effects from Zhang & Bareinboim (2018) as “path-specific causal fairness”.

Action: We have clarified throughout the paper that our framework is general and can also be applied to other fairness notations. We also added new theoretical results for other fairness notions (see our new Supplement F).

3) Extended literature review:

Thank you. We extended our literature overview and now provide a comprehensive review of prior literature on sensitivity analysis on causal effects to unobserved confounding, such as the marginal sensitivity model. In particular, we added the following materials:

• We added a comprehensive literature overview on sensitivity analysis (see our new Supplement C.2). Therein, we review existing works but also discuss that these are mainly focused on causal effects and not causal fairness. To the best of our knowledge, there is no work other than ours that has combined sensitivity analysis and causal fairness. As we lay out, our paper is the first to make predictions wrt. causal fairness under unobserved confounding.
• We added a discussion of the strengths and weaknesses of various sensitivity models. Thereby, we highlight the reasons for adopting the GMSM in our work (see our new Supplement C.2).

4) Enhanced experimental analysis:

Thank you. We are more than happy to expand our experimental analysis. We thus included several new experimental analyses as follows:

• We report the prediction performance in Supplement G. (Due to space constraints, we do so in the supplements and not in the main paper.)
• We further show the robustness of our framework. For this, we added additional experimental results on the performance of our fair classifier when trained with different sensitivity parameters (see our new Supplement I).

5) Continuous variables:

Thank you. We improved our paper to show that our framework is also effective for continuous variables:

• We clarified that the features (confounders) are not restricted to discrete variables. Instead, we now explain that our framework is applicable to multiple features that can be binary, continuous, and/or discrete. Therefore, our method is also applicable to multiple continuous and, thus as well to high-dimensional features (see our revised Section 3).
• We provide new experimental results to demonstrate the effectiveness of our framework for multiple continuous variables (see new Supplement J).
• We also discussed an extension of our framework for continuous mediators (see new Supplement J).

Response to question

Response to “Benefits of the GMSM”

Thank you for bringing up this important question. We added a discussion of the strengths and weaknesses of various sensitivity models. Thereby, we highlight the reasons for adopting the GMSM in our work (see our new Supplement C.2).

I thank the authors for responding to all the comments, which addresses some of my concern.

I acknowledge that effectively implementing the sensitivity model necessitates the simplification of causal fairness notions, as defined in Pearl's framework, from level 3 to level 2. This process indeed requires amount of work and careful consideration.

However, as Reviewer rYvN propose, the equation and algorithms is still unclear. This work can be improved from many aspects. To be more clear, the paper structure can be further improved. I hope my comments are helpful for the revision.

Thanks for the authors' careful responding to all the comments. I will raise my score.

However, I still think it is not suitable to claim the contributions on causal fairness in the main body (and title), since the paper actually is confined to specific causality-based fairness notions based on counterfactual direct effect (Ctf-DE), indirect effect (Ctf-IE) and spurious effects, instead of the general causal fairness notions. Although the authors added a new Supplement F, deriving bounds for fair on average causal effect and a simple path-specific causal fairness, it does not encompass all causal fairness notions, like counterfactual fairness or path-specific counterfactual fairness. It seems the authors mix the use of some causal fairness notions sometime. For example, it is also confusing to change 'path-specific causal effect' to 'path-specific causal fairness', since actually path-specific causal fairness is defined on path-specific causal effect and they are the same thing in the essence. To avoid confusion and overclaim, I think it is better to reorganize the structure of the paper.

In conclusion, I acknowledge the importance of the performing sensitivity analysis in the causal fairness area, however I still do not think this paper is ready to be published and once again remark that I hope to see an improved version of your work published in the future.

Thank you for your positive evaluation of our paper! We took all your comments at heart and improved our paper accordingly.

Response to (1): Connection of motivation and solution

We thank the reviewer for the question and are happy to clarify the connection between our motivation and the proposed solution. In practice, knowledge of the unobserved confounding (UC) strength is beneficial but not completely necessary. Importantly, our framework is of direct help in practice when (1.) the maximum UC strength is known or (2.) unknown. We discuss both use cases below:

1. There are often good reasons why the UC strength is known or why it can be upper-bounded. Many fairness-relevant applications are rooted in social science, where there is a good understanding of potential causes for biases (e.g., around gender, age, etc.). Hence, it is often reasonable to make assumptions that the UC strength is in a specific relationship with some other observed variable [1, 2, 3, 4], e.g., not as strong as (a multiple of) the observed variable. Notwithstanding, several approaches have been proposed on how sensitivity parameters can be chosen in practice. Here, we refer to the discussions in [5, 6, 7, 8].
2. Even when the UC strength is unknown, our framework is of significant value in practice. In this case, practitioners can employ our bounds to audit the unobserved confounding in the data. To do so, one can use our framework to calculate bounds for increasing sensitivity parameters until the interval defined by the bounds contains a certain value of interest, e.g., zero (indicating complete fairness). The minimal sensitivity parameter to achieve this goal corresponds to the level of unobserved confounding, which would be necessary to consider the data fair. We can also view the sensitivity parameter as our uncertainty about the fairness of our prediction model. Smaller sensitivity parameters not achieving the goal still provide information about the sign, i.e., direction, of the unfair effect [6]. Generally, testing various sensitivity parameters is a common technique in real-world applications of sensitivity analysis [5, 9, 10]. In sum, even when the UC strength is unknown, our framework can thus be of large practical value.

Actions: We have carefully revised our paper in the following ways:

• We have improved our motivation to clarify the connection to our solution framework (see revised introduction).
• We added a discussion on the practical considerations from above (see new Supplement B.2).

Response to (2): Eq. (11)

Thank you. We have now improved the notation in Eq. (11). Note that the prediction model returns class probabilities for classification or the delta distribution (point distribution) for regression tasks. Therefore, $P(Y)$ does not correspond to the true distribution of the outcome variable $Y$ but rather to the conditional distribution over the prediction outcomes $P(\hat{Y} \mid z, m, a)$, where $\hat{Y} \mid z, m, a$ defines the prediction outcome, i.e., the predicted distribution of $Y$, for a given realization of $Z, M$ and $A$. To clarify this, we have updated our notation accordingly and provided further clarifications in the text.

Taking the expectation over the class probabilities or the delta distribution is a natural approach in binary classification and regression problems. For multiclass classification, other distribution quantiles are separate constraints per output class might also be reasonable depending on the specific requirements of the use case.

Action: We have revised Eq. (11) along with the suggestions in your comment.

Response to (3): Sensitivity analysis between $A$ and $Y$

Thank you for your question. Indeed, it is not necessary to perform sensitivity analysis on the relationship between $A$ and $Y$ since the confounding on the relationship vanishes by design. This means that we do not need to define a sensitivity model for $Y$, i.e., we do not need to specify $\Gamma_Y$. Nevertheless, we still need a sensitivity model for $M$ to obtain bounds $P^+(m \mid a, z), P^-(m \mid a, z)$ for $P(m \mid a, z)$.

The sensitivity models that need to be specified do not directly map to the three path-specific causal fairness effects (direct, indirect, spurious). They only provide bounds on a given probability. Since all three effects contain $P(m \mid a, z)$, we obtain bounds for all effects, not only for the indirect effect.

If we want to assess the dataset fairness, we need to bound $P(y \mid a, z, m)$ and $P(m \mid a, z)$. Therefore, in this case, we need to specify two sensitivity models, i.e. one with $\Gamma_M$ and one with $\Gamma_Y$.

Response to (4): Eq. (33)

Thank you for the suggestion on how to improve our notation (previous: Eq. 33; now Eq. 32). We did not intend to condition on measure-zero events but to intervene on conditional events, i.e., set the sensitive attribute to $a_j$ by intervention for events in which the realization was $a_i$. We have thus revised our notation. Reassuringly, we emphasize that all subsequent applications of the equation were correct.

Action: We have rewritten the equation (now: Eq. 32, previous: Eq. 33) in terms of counterfactual expressions to avoid misleading notation.

Response to (5): Applicability to other path-specific effects

Thank you. Our framework is specially designed to incorporate path-specific effects. Nevertheless, we want to emphasize that our framework is general and can also be applied to other fairness notions. To demonstrate that, we added a new Supplement F, where we derive bounds for other notions. Thereby, we demonstrate how our framework can be applied to further fairness notions, which shows the broad applicability of our framework.

Action: We have clarified throughout the paper that our framework is general and can also be applied to other fairness notations. We also added new theoretical results for other fairness notions (see our new Supplement F).

Thank you for your very constructive and detailed review of our paper! We took all your comments at heart and improved our paper accordingly.

Response to (1): Difference from Frauen et al (2023)

Thank you for giving us the opportunity to emphasize the novelty of our framework. Thereby, we clarify that we do not have a simple application of Frauen et al. (2023) but that our derivations are non-trivial.

Why is the derivation of our bounds non-trivial? We have derived tailored bounds for counterfactual fairness under unobserved confounding. Importantly, our bounds require a novel and careful derivation. In other words, our bounds in Theorem 1 are not just a simple result of applying Frauen et al. (2023) but require a new derivation tailored to our setting. The reason is the following: Causal fairness notions commonly contain nested counterfactuals and are thus located on level three of Pearl’s ladder of causality. The GMSM in Frauen et al. (2023) incorporates single-intervention counterfactuals and is thus located on level two. Because of that, existing bounds from sensitivity models are not applicable. In other words, we can not simply adopt existing bounds from sensitivity models, but we need a new derivation for level three of Pearl’s causality ladder.

What is our contribution? We use the GMSM from Frauen et al (2023) primarily to formalize our setting, while the actual derivation for our bounds is new. Hence, our derivation does not simply follow from the GMSM but is non-trival (i.e., we only use the GMSM from Lemma 1 to formalize our setting while our Theorem 1 and its derivation is new). Our main contributions are beyond the GMSM. (1) We transfer concepts from sensitivity analysis to causal fairness literature. This is a major difference from the existing literature on causal fairness, which has ignored the fairness implications due to unobserved confounding. As such, we fill an important research gap in the literature (see our new Table 1). (2) We propose a new neural framework to assess causal fairness under unobserved confounding and learn robust predictions. Our is the first framework to mitigate fairness violations from unobserved confounding so that still fair predictions are made. (3) We offer new insights for practice: one of our implications is that there can be important fairness uses in practice due to unobserved confounding, because of which fairness in practice can be undermined. We thus expect that our findings, as well as our framework, are of immediate practical importance.

How is our framework novel compared to sensitivity analysis? While we adopt a sensitivity analysis as the basis for deriving our bound, we add over sensitivity analysis in several ways. (1) We derive new bounds that are tailored to our bounds. The bounds do not directly follow from the sensitivity analysis (which is unlike other applications of sensitivity models in the literature); instead, a tailored derivation is needed that is non-trivial. (2) We develop an end-to-end framework for fair prediction models. As such, Steps 1 and 2 in our framework are novel and make large contributions to existing sensitivity analysis.

Actions: We improved our work in the following ways to clarify our novelty and why our contributions are non-trivial:

1. We spell out clearly that our setting is grounded in Frauen et al. However, our derivations are new and non-trivial (see revised Section 2). We further explain why existing bounds from Frauen et al. are not applicable (as our causal query is in ladder three and not in ladder two of Pearl’s causality ladder).
2. We added a technical background explaining the theoretical differences between existing bounds from sensitivity models (=ladder two) and our causal query (=ladder three). Thereby, we motivate why a new derivation is needed (see new Supplement C.3).
3. We included a new Table 1, where we highlight that our work is orthogonal to sensitivity analysis and that we make important contributions.

Thank you, while I do think that my statement that the main idea is the application of Lemma 1 to the existing bounds of GMSM was correct (Remark 1 is basically the main idea, right?), I acknowledge that by looking at the derivation in the appendix in more detail I agree that it still required quite some work. Even digesting and presenting Correa et al. into this context is a non-trivial effort, which I recognize.

It's frustrating that the original pdf is not available anymore (or maybe I'm just incompetent in trying to find it), but wasn't Eq. 11 just $\mathbb E[Y]$ originally? My main point was that averaging over $A$ and $M$ doesn't seem to make sense to me (imagine that we want to optimize some $\mathbb E[f(Y, A, M)]$ where we need $f(y, a, m)$ to be positive for all $a$, $m$, but we only enforce $\mathbb E[f(Y, A, M)]$ to be positive...). The "max" in the new Eq. 11 is ambiguous. Can you clarify what is this a maximum over what? All $m, a$? (across $CF+$ and $CF-$). I'm not sure whether the iterations in Algorithm 1 are doing that.

Thank you for seeing our paper in a good light. We welcome your follow-up questions and are more than happy to answer them below. We acknowledge that our notation was ambiguous ($Y\mid m,a,z$) and hope that our responses and updates in our manuscript can address your questions and concerns successfully. Specifically, we introduced the notation of expected fairness effects ($\mathrm{CF}\_{\mathbb{E}}$) to clarify the notation.

General clarifications: All fairness effects are defined for specific realizations of $Y$ and $A$, e.g., $DE_{a_i, a_j}(y|a_i)$. For calculating the bounds, we replace $P(y|m,a,z)$ through our prediction outcome for $f(m,a,z)$. Although $P(y|m,a,z)$ is a random variable (depending on $y$) for deriving standard fairness bounds when training the prediction model, it is a non-random variable, independent of $y$, i.e., $f(m,a,z)$. Then, following the definition of the bounds given in Thm.1, we average over $M$ and $Z$ and finally obtain a non-random quantity that only depends on the realization of $A$. We denote these quantities as $\mathrm{CF}\_{\mathbb{E}}^+(a_i, a_j)$ and $\mathrm{CF}\_{\mathbb{E}}^-(a_i, a_j)$ for $\mathrm{CF} \in {\mathrm{DE}, \mathrm{IE}, \mathrm{SE}}$. We employ a notation containing the expectation, to explicitly show the non-randomness and the independence of $y$.

In the following, we outline further details on the derivations for each of the tasks binary classification, regression and mutli-class classification separately. We will switch from the notation used in our manuscript to the notation used in your previous question, i.e. random variables $X$,$Y$ below.

First, let’s take a look at binary variables $X,Y$, which also corresponds to our implementation in the experiments on synthetic data. For each of the three different fairness effects, we are then only interested in, e.g., $\mathrm{CF} (Y=1 \mid X=1)$. Due to symmetry reasons for binary $X$ (e.g., discrimination of women compared to men is symmetric to discrimination of men compared to women), it is sufficient to focus on one realization of $X$. In our case, we thus have one constraint per $\mathrm{CF}$, i.e., three constraints and thus three Lagrange multipliers in total.

Next, let us consider the regression setting, i.e., $Y$ continuous, $X$ binary, which corresponds to our real-world study. Here, it is reasonable to consider the expectation over $Y$. We are thus left again with three constraints in total (one per $\mathrm{CF}$ ).

Finally, let’s consider multi-class classification. Now there are two options of defining constraints to optimize over. One option is (as you noted in your question) to treat each $\mathrm{CF}$ for each possible realization of $Y$ and thus minimize wrt. $3 \times |\mathcal{Y}|$ constraints. Although this is feasible for binary classification, it can be highly challenging for large $|\mathcal{Y}|$. Therefore, an alternative option is to constrain the expected $\mathrm{CF}$ over all realizations of $Y$. The particular choice of which notion to constrain is up to the practitioner in real-world use cases. We added a clarification regarding this choice in our updated manuscript and updated Algorithm 1 to spell this out more explicitly.

The max function is then applied following Eq. (11), i.e., on each constraint separately (=which is also what you stated in your question). Each max function only takes the maximum of the absolute value of the upper or lower bound of one specific $\mathrm{CF}$ (either averaged over all outcomes of $Y$ or one per outcome separately). Therefore, there are # constraints-many max functions and, thus, # constraints-many Lagrange multipliers. Overall, $\lambda, c$ and $\gamma$ in the pseudo-code are vectors of dimension # constraints.

The function $max(x,y)$ is differentiable unless $x=y$. In this case, one can observe that the subdifferential of $max(x,y)$ at $(x,y)$ is given by the convex hull of $[(1,0), (0,1)]$. In general, choosing any subgradient at $(x,y)$ circumvents the differentiability problems. To furthermore ensure computational stability, each of the gradients wrt. $x$ or $y$ is best set to 0.5, i.e., the element in $[(1,0), (0,1)]$ with the smallest norm. Common deep learning libraries directly implement this rule in the provided functions.

Action: Upon reading your question, we realized that we should clarify the algorithm in our paper. To this end, we have updated Algorithm 1 in our paper and added a further discussion in Supplement E.4. In our updated algorithm, we now state the different multipliers more explicitly. Furthermore, we have now specified the max operation in line 6 more rigorously.

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

Thank you again very much for the helpful discussion on the mathematical representation of our framework. We highly value your comments and input, which have significantly improved the presentation in our paper.

We will carefully revise the paper following your suggestions and make sure to fully spell out all changes between whole functions and expected values. In particular, for an eventual camera-ready version, we will revise the complete notation in our paper again to ensure consistency. If you have any additional remarks or questions about the notation, we are more than happy to incorporate them in a revised version of the paper.

We also have released the code in an anonymous GitHub repository (https://anonymous.4open.science/r/FairSensitivityAnalysis_anonymous-0DA2/README.md / the link is also in the paper) to preserve anonymity. Upon acceptance, we will move our codes to a public GitHub.

We want to thank reviewer rYvN for noting the ambiguity of our notation in the presentation of Algorithm 1. Based on the reviewer’s valuable suggestions, we improved the presentation in the following way:

• We improved the notation for our constraints in Section 5 and Algorithm 1 and added further explanations on the calculations performed in our framework.
• We present an additional alternative training algorithm (see our new Algorithm 2) for multi-class classification in our new Supplement E.4 along with a discussion of both algorithms applied to different tasks, i.e., binary classification, mutli-class classification, and regression.

We added the new changes in red color.