Paper: Causal Fairness under Unobserved Confounding: A Neural Sensitivity Framework (ICLR-2024)

Reviewer comment (Theory):
Limited Scope in Fairness Notions. The paper is somewhat misleading in its claim to perform sensitivity analysis on causal fairness under unobserved confounding. In fact, the focus of this paper is confined to specific causality-based fairness notions based on counterfactual direct effect (Ctf-DE), indirect effect (Ctf-IE) and spurious effects. However, causal fairness encompasses a broader range of notions, such as ones based on proxy discrimination, path-specific causal effects, path-specific counterfactual effects (including natural direct or indirect causal effect), etc. Additionally, it is misunderstanding to call the definitions Ctf-DE, Ctf-IE and Ctf-SE in Zhang & Bareinboim (2018a) as ‘path-specific causal effects’, which differs from their formal definition [1], thus leading to confusion.

Author response:
**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**).

Reviewer comment (Novelty):
Incomplete Literature Review. The paper lacks a comprehensive review of prior literature on sensitivity analysis on causal effects to unobserved confounding, such as marginal sensitivity model. Additionally, it would be beneficial to provide a brief overview of sensitivity analysis models, including the GMSM, and discuss their strengths and weaknesses.

Author response:
**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**).

Reviewer comment (Experiments):
Enhanced Experimental Analysis. In the experimental section, it would be advantageous to report and compare prediction performance across various models under different levels of confounding. This would provide a more robust assessment of the proposed method's performance.

Author response:
**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**).

Reviewer comment (Novelty):
The authors state that "A key benefit of the GMSM is that it can deal with discrete mediators and both discrete and continuous outcomes." It is not convincing. It would be helpful to elaborate on other reasons for adopting the GMSM and discuss the strengths and weaknesses of alternative sensitivity models.

Author response:
**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**).

Reviewer comment (Theory):
The motivation and solution seem disconnected. Authors suggest practitioners can audit the unobserved confounding in the data in the introduction, but the solution using GMSM requires prior knowledge of UC strength. In this case, we need to know the unobserved confounding strength in order to estimate or optimize the causal fairness.

Author response:
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**).

Reviewer comment (Evaluation):
Authors did not report accuracy measures in the paper, which should also be reported. I assume such fairness constraint would impact the prediction accuracy a lot.

Author response:
Thank you. We are more than happy to report the prediction performance in **Supplement G**.

Reviewer comment (Experiments):
A natural criticism is the method is restricted to discrete features, while in practice, variables like crime history may be highly contextual (videos, images, text, ..). Can authors discuss the difficulty in extending it to high dimensional features?

Author response:
Thank you. We improved our paper to show that our framework is also effective for high-dimensional features: 
* 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 as 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**).

Reviewer comment (Theory):
Can authors explain more on "Fairness bounds in non-identifiable settings": why general non-identifiability in these papers does not encompass the unobserved confounding case? Nabi & Shpitser, 2018 also uses unobserved confounding as one of the examples.

Author response:
Thank you. We appreciate the opportunity to clarify the **difference** between our framework for unobserved confounding and existing work on general non-identifiability. Of note, both are **different** concepts. Unobserved confounding is only one reason for non-identifiability. Other reasons are, e.g., interference or lack of overlap. Methods able to provide fair predictions in general non-identifiable settings are thus not tailored to unobserved confounding specifically, resulting, e.g., in inefficient fairness constraints.


The paper by Nabi & Shpitser [11] provides an excellent motivation for our paper. Therein, the authors state that causal effects are unidentifiable in the presence of unobserved confounding, and, as a result, causal fairness may be violated due to unobserved confounding. However, the paper by Nabi & Shpitser [11] provides merely a theoretical conceptualization of the problem but **without an actual algorithm** to solve the problem. The latter (i.e., an algorithm to mitigate issues from unobserved confounding for casual fairness) is our novelty. Because of that, Nabi & Shpitser [11] is also **not** applicable as a baseline. In sum, the paper by Nabi & Shpitser is **vastly different from ours**: their paper does **not** quantify the impact on fairness from unobserved confounding, and it does mitigate fairness issues due to unobserved confounding. In contrast, only our paper **provides fairness bounds** and **a framework** for training fair prediction models in the presence of unobserved confounding.

Reviewer comment (Writing):
Typo: abstract, sources of unobserved confounding? "ours is the first work to study causal fairness under observed confounding." miss-specification.

Author response:
Thank you. We fixed the text accordingly (see blue text in Section 1).

Reviewer comment (Writing):
Writing: Why repeat the contribution at the end of section 1. The research gap seems to repeat contribution again and such gap is already mentioned before.

Author response:
Thank you. We fixed the text accordingly (see blue text in Section 1).

Reviewer comment (Novelty):
My main comment is to what extent this mostly is a direct application of Frauen et al. (2023). If I understood it well, is the main technical innovation the use of Lemma 1 in the context of GMSM? Or is this even more closely related to the cited paper?

Author response:
**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.

Reviewer comment (Theory):
I’m confused by Eq. (11): there is an expectation over $Y$. Why is this form of aggregation sensible? I’d expect that we would like to enforce the constraint uniformly over all possible values in the same sample space of $Y$ instead of performing any sort of probabilistic averaging. Speaking of which, which distribution are we marginalizing over, is this the marginal distribution of $Y$? What happens to $a_i$, $a_j$ from (1)–(3)?

Author response:
**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.

Reviewer comment (Theory):
Maybe this is well-explored in the causal fairness literature, but the switch between “real Y” and $f_\theta$ in the training procedure in Section 5 is very confusing: we design $f$, so any confounding between “$Y$” (that is, $f_\theta$) vanishes by design. Why would we consider any sort of sensitivity parameter for “$Y$”?

Author response:
**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 $\Gamma_Y$.

Reviewer comment (Theory):
I’m worried about the theoretical results. In Appendix D.3 the authors state that they “extensively use the following corollary”, which boils down to Eq. (33). This equation refers to “$P(X |do(A=a), A\neq a)$”. The event past the conditioning bar has probability zero, as the control signal $do(A = a)$ implies $A = a$ (yes, we can condition on measure-zero events, that’s what we do for continuous variables, but even then we take for granted the careful textbook measure-theoretical characterization of this conditional that we implicitly accept by default, but that’s a non-trivial (and not unique!) characterization of conditioning. But (33) doesn’t appear to make sense even for discrete $A$). I may be missing something obvious, in which case I will appreciate a clarification – if this indeed “follows directly from basic probability theory” I’m happy to be taught the baby steps explicitly. Of course we can say that 

$$P(X | do(A = a)) = P(X(a)) = P(X(a) | A = a)P(A = a) + P(X(a) | A \neq a)P(A \neq a) = $$
$$P(X | A = a)P(A = a) + P(X(a) | A \neq a)P(A \neq a) , $$

but $P(X(a) | A \neq a)$ is not the same thing as $P(X | do(A = a), A \neq a)$. Among other things, $X$ is not $X(a)$ when $A \neq a$, but it is undefined what $X$ even means under regime $do(A = a)$ and event $A \neq a$.

Author response:
**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.