Causal Context Connects Counterfactual Fairness to Robust Prediction and Group Fairness
Authors: Jacy Reese Anthis
Victor Veitch
Venue: NeurIPS 2023 poster (2023)
Keywords: causal graphs
causality
counterfactual fairness
domain generalization
fairness
robustness
machine learning
artificial intelligence
TL;DR: We motivate counterfactual fairness via out-of-distribution accuracy and develop a correspondence between counterfactual fairness and group fairness metrics.

Abstract: Counterfactual fairness requires that a person would have been classified in the same way by an AI or other algorithmic system if they had a different protected class, such as a different race or gender. This is an intuitive standard, as reflected in the U.S. legal system, but its use is limited because counterfactuals cannot be directly observed in real-world data. On the other hand, group fairness metrics (e.g., demographic parity or equalized odds) are less intuitive but more readily observed. In this paper, we use \textit{causal context} to bridge the gaps between counterfactual fairness, robust prediction, and group fairness. First, we motivate counterfactual fairness by showing that there is not necessarily a fundamental trade-off between fairness and accuracy because, under plausible conditions, the counterfactually fair predictor is in fact accuracy-optimal in an unbiased target distribution. Second, we develop a correspondence between the causal graph of the data-generating process and which, if any, group fairness metrics are equivalent to counterfactual fairness. Third, we show that in three common fairness contexts—measurement error, selection on label, and selection on predictors—counterfactual fairness is equivalent to demographic parity, equalized odds, and calibration, respectively. Counterfactual fairness can sometimes be tested by measuring relatively simple group fairness metrics.

Causal Context Connects Counterfactual Fairness to Robust Prediction and Group Fairness
Jacy Reese Anthis Victor Veitch' University of Chicago, University of California, Berkeley, "Sentience Institute
Abstract
Counterfactual fairness requires that person would have been classified in the same way by an AI or other algorithmic system they had a different protected class, such as a different race or gender. This is intuitive standard, reflected in the U.S. legal system, but its use is limited because counterfactuals cannot be directly observed in real-world data. On the other hand, group fairness metrics (e.g., demographic parity or equalized odds) are less intuitive but more readily observed. causal context to bridge the gaps between counterfactual
In this paper, fairness, robust prediction, and group fairness. First, we motivate counterfactual fairness by showing that there is not necessarily fundamental trade-off between fairness and accuracy because, under plausible conditions, the counterfactually fair predictor is in fact accuracy-optimal in an unbiased target distribution. Second, we develop a correspondence between the causal graph of the data-generating process and which, if any, group fairness metrics are equivalent to counterfactual fairness. Third, we show that in three common fairness menexxts-massuemmment error, selection on label, and selection on predictors counterfactual fairness is equivalent to demographic parity, equalized odds, and calibration, respectively. Counterfactual fairness can sometimes be tested by measuring relatively simple group fairness metrics.
Introduction
The increasing use of artificial intelligence and machine learning in high stakes contexts such as healthcare, hiring, and financial lending has driven widespread interest in algorithmic fairness. A canonical example is risk assessment in the U.S. judicial system, which became well-known after 2016 investigation into the recidivism prediction tool COMPAS revealed significant racial disparities (2. There are several metries one can use to operationalize fairness. These typically refer protected class or sensitive label, such as race or gender, and define an equality of prediction rates across the protected class. For example, demographic parity, also known statistical parity or group parity, requires equal classification rates across all levels of the protected class 7
However, there is open challenge of deciding which metrics to enforce in given context [e.g.. 24 44). Appropriateness can vary based on different ways to measure false positive and false negative rates and the costs of such emrrr [19) well as the potential trade-off's between fairness and accuracy
[2 21 55 Moreover, there are well-known technical results showing that it is impossible to
achieve the different fairness metries simultaneously, even c-approximation [10] (30. which suggest that a practitioner's context-specific perspective may be necessary to achieve satisfactory outcomes
different paradigm, counterfactual fairness, requires that a prediction would have been the same the person had different protected class (31). This matches common intuitions and legal standards
Corresponding author: anthis @ @wuc@uuuhhhiccuduu
37th Conference on Neural Information Processing Systems (NeurIPS 2023).

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Tools to Detect and Enforce Group Fairness
Causal Context Connection
Group Fairness Metries
Counterfactual Fairness
Data augmentation
Demographic parity
Theorem
Ethics philosophy
Data generation
Equalized odds
Corollary 2.1
Public policy
Regularization
Calibration
Corollary 2.2
Law
Etc.
Etc.
Etc.
Figure 1: A pipeline detect and enforce counterfactual fairness. This facilitated by [hhorem 2 correspondence between group faimess metrics and counterfactual faimess, such that tools to detect and enforce group fairness can be applied to counterfactual fairness.
25). For example, in McCleskey Kemp (1987), the Supreme Court found "disproportionate impact" as shown by group disparity to be insufficient evidence of discrimination and required evidence that "racial considerations played part." The Court found that there the Georgia legislature had violated the Equal Protection Clause via its use of capital punishment on the grounds that it had not been shown that this was [sic] "because of anticipated racially discriminatory effect." While judges do not formally specify causal models, their language usually evokes such counterfactuals; see, for example, the of "but-for causation" in Bostock v. Clayton County (2020).
merit the argument that
We take new perspective counterfactual fairness by leveraging causal context. First, provide a novel motivation for counterfactual fairness from the perspective robust prediction by showing that, with certain causal structures of the data-generating process, counterfactually fair predictors accuracy-optimal in an unbiased target distribution (Thorem Tp. The contexts we consider ihave two important features: the faithfulness of the causal structure, in the sense that causal effects happen to be precisely counterbalanced, which could lead to coincidental achievement of group fairness metrics, and that the association between the label Y and the protected class A is "purely spurious," in the sense that intervening on A does not affect or vice versa.
Second, address the fundamental challenge in enforcing counterfactual fairness, which is that, by definition, we never directly observe counterfactual information in the real world. To deal with this, show that the causal context connects counterfactual faimness observational group fairness metrics (Theorem 2 and Corollary 2. and this correspondence can be used to apply tools from group fairness frameworks to the ideal of achieving counterfactual fairness (Figure []. For example, the fairness literature has developed techniques to achieve group fairness through augmenting the input data [3 27 data generation [T], or regularization [45). With this pipeline, these tools can be applied to enforce counterfactual fairness by achieving the specific group fairness metric that corresponds counterfactual fairness in the given context. In particular, we show that in each fairness context shown in Figure 2 counterfactual fairness is equivalent to a particular metric (Corollary 2.2). This correspondence can be used to apply tools built for group fairness the goal of counterfactual fairness or vice versa.
Finally, conduct brief experiments in semi-synthetic setting with the Adult income dataset to confirm that a counterfactually fair predictor under these conditions achieves out-of-distribution accuracy and the corresponding group faimess metric. To do this, develop novel counterfactually fair predictor that is a weighted average of naive predictors, each under the assumption that the observation is in each protected class (Theorem 3).

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x
Y
x
A
X
A
X
(a) Mcasurement error (corresponding demographic parity)
(b) Selection on label (corresponding to equalized odds)
S
A
Xy
(c) Selection on predictors (corresponding to calibration)
Figure 2: DAGs for three causal contexts which counterfactually fair predictor is equivalent particular group fairness metric. Measurement error represented with the unobserved true label Y in dashed circle, which is parent of the observed label Y, and selection is represented with the selection status S in a rectangle, indicating an observed variable that conditioned on in the data-generating process, which induces an association between its parents. Bidirectional arrows indicate that either variable could affect the other.
In summary, we make three main contributions:
We provide a novel motivation for counterfactual fairness from the perspective of robust prediction by showing that, in certain causal contexts, the counterfactually fair predictor is sccuracy-optimal an unbiased target distribution (Theorem I).
2. We provide criteria for whether a causal context implies equivalence between counterfactual faireess and each of the three primary group fairness metrics (Theorem as well as seven derivative metries (Corollary 2.
3. We provide three causal contexts, shown in Figure 2 is equivalent to a particular group fairness metric: measurement error with demographic parity, selection on label with equalized odds, and selection on predictors with calibration (Corollary 2.2).
which counterfactual fairness
For running example, consider the case of recidivism prediction with measurement error, as shown in Figure 2a where X is the set of predictors of recidivism, such as past convictions; Y is committing crime; Y is committing a crime that has been reported and prosecuted; and A is race, which in this example affects the likelihood crime is reported and prosecuted but not the likelihood of crime itself. In this case, we show that predictor is counterfactually fair and only if it achieves demographic parity.
2 Related work
Previous work developed number of fairness metrics, particularly demographic parity 7 ized odds [23], and calibration [19]. Verma and Rubin [48] details 15 group fairness metrics alsrics-also known as observational or distributional fairness metrics metrics-including these three. and Makhlouf, Zhioua, and Palamidessi 34 details 19 causal faimess metries, including counterfactual fairness 31.. Many notions of fairnsss can be viewed as specific instances the general goal of invariant representation [18
equal-
53 55 56). Much contemporary work in algorithmic fairness develops ways to better enforce certain faimess metrics, such as through reprogramming a pretrained model [54].

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contrastive learning when most data lacks labels of the protected class [8. and differentiating between critical and non-critical features for selective removal [[6].
Several papers have criticized the group fairness metrics and enriched them correspondingly. For example, one can achieve demographic parity by arbitrarily classifying a subset of individuals in a protected class, regardless of the other characteristics of those individuals. This is particularly important for subgroup fairness, in which predictor can achieve the fairness metric for one category within the protected class but fail to achieve it when subgroups are considered across different protected classes, such as people who have certain race and gender. This is particularly important given the intersectional nature of social disparities [14), and metrics have been modified to include possible subgroups [29).
Well-known impossibility theorems show that fairness metrics such as the three listed above cannot be simultaneously enforced outside of unusual situations such perfectly accurate classifier, random classifier, or a population with equal rates across the protected class, and this is true even to an e-approximation [10] 30 40). Moreover, insofar as fairness and classification accuracy diverge, there appears to be a trade-off between the two objectives. Two well-known papers in the literature, Corbett-Davies et al. [12] and Corbett-Davies and Goel (T.. detail the cost of faimess and argue for
the prioritization of c classification accuracy. More recent work on this trade-off includes a geometric characterization and Pareto optimal frontier for invariance goals such as fairesss (55), the cost of causal fairness in terms of Pareto dominance in classification accuracy and diversity (37), the potential for increases in fairness via data reweighting in some cases without a cost classification accuracy [2], and showing optimal fairness and accuracy in an ideal distribution given mismatched
distributional data [17.
Our project uses the framework of causality to enrich fairness metric selection. Recent work has used
structural causal models to formalize and solve a wide range of causal problems in machine learning,
such as disentangled representations [41 (50. -ut-of-distribution generalization [41] 49 50), and the
identification and removal of spurious correlations [47] [5). While Veitch et al. [47] focuses on the
correspondence between counterfactual invariance and domain generalization, Remark 3.3 notes that counterfactual fairness as an instance of counterfactual invariance can imply either demographic parity or equalized odds in the two graphs they analyze, respectively, and Makar and D' 'Amour (3 draw the correspondence between risk invariance and equalized odds under a certain graph. The present work can be viewed as generalization of these results to correspondence between counterfactual fairness, which equivalent to risk invariance when the association is "purely spurious," " and each of the group fairness metrics.
3 Preliminaries and Examples
For simplicity of exposition, we focus on binary classification with an input dataset X E X and binary label Y y {0, 1), and in our data we may have ground-truth unobserved labels y E {0. that do not always match the potentially noisy observed labels Y. The goal is to estimate a prediction function f X' + y, producing estimated labels s(X) E (0. 1 1). that minimizes E[E(f(X) Y)] for some loss function É y x y + I. There are number of popular fairness metrics in the literature that can be formulated as conditional probabilities for some protected class A E {0, 1}, or equivalently, independence of random variables, and we can define notion of counterfactual fairness.
Definition 1. (Demographic parity). A predictor f(X) achieves demographic parity if and only if: P(f(X)
A 0) ((f(X) = 1 = f(X) LA
Definition 2. (Equalized odds). A predictor f(X) achieves equalized odds if and only if:
P((()= = 1 A 0, Y = y) = ((f(X) - - 1, = < S(X) T A AlY Y
Definition 3. (Calibration). predictor f(X) achieves calibration if and only if:
I(Y A = 0, f(X) = 1) = P(Y A Definition& 4. (Counterfactual fairness). A predictor f(X) achieves counterfactual fairness if and only if for any a, a' A:
A = 1,, 1) - T f(X)
f(X(a)) = f(((aa))
4

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In this case, z00) and z(1) are the potential outcomes. That is, for an individual associated with data x E X with protected class a E A, their counterfactual is z(a') where a' E A and a * a'. The counterfactual observation is what would obtain if the individual had a different protected class. We represent causal structures as directed acyelic graphs (DAGs) in which nodes are variables and directed arrows indicates the direction of causal effect. Nodes in a solid circle are observed variables that may or may not be included in a predictive model. A dashed circle indicates an unobserved variable, such as in the case of measurement error in which the true label Y a parent of the observed label Y. A rectangle indicates an observed variable that is conditioned on in the data-generating process, typically denoted by S for a selection effect, which induces an association between its parents. For clarity, we decompose X into X . the component that is not causally affected by A. and Xý, the component that does not causally affect Y (in causal-direction graph, i.e., X affects Y) or is not causally affected by Y (in an anti-causal graph, i.e., affects X). In general, there could be a third component that is causally connected to both A and Y (i.e., a non-spurious interaction effect). This third component is excluded through the assumption that the association between Y and A in the training distribution is purely spurious [47].
Definition 5. (Purely spurious). We say the association between Y and A is purely spurious if Y A.
In other words, if we condition on A, then the only information about of the input that is not causally affected by the protected class). We also restrict ourselves to cases in which A is exogenous to the causal structure (i.e., there is no confounder of A and X or of A and Y), which seems reasonable in the case of fairesss because the protected class is typically not caused by other variables in the specified model.
X (i.e., the component
Despite the intuitive appeal and legal precedent for counterfactual fairness, the determination of counterfactuals fundamentally contentious because, by defimition, we do not observe counterfactual worlds, and we cannot run scientific experiments to test what would have happened in different world history. Some counterfactuals are relatively clear, but counterfactuals in contexts of protected classes such as race, gender, and disability can be ambiguous. Consider Bob, hypothetical Black criminal defendant being assessed for recidivism who is determined to high-risk by human judge or machine algorithm. The assessment involves extensive information about his life: where he has lived, how he has been employed, what crimes he has been arrested and convicted of before, and so on. What is the counterfactual in which Bob is White? Is it the world in which Bob was born to White parents, the one in which Bob's parents were themselves born to White parents, or another change further back? Would Bob still have the same educational and economic circumstances? Would those White parents have raised Bob in the same majority-B -Back neighborhood he grew up in or in majority- White neighborhood? Questions of counterfactual ambiguity do not have established answers in the fairness literature, either in philosophy (35). epidemiology [46), or the nascent machine learning literature (8 and we do not attempt to resolve them in the present work.
Additionally, defining counterfactual fairness in given context requires some identification of the causal structure of the data generating process. Fortunately, mitigating this challenge is more technically tractable than mitigating ambiguity, as there is an extensive literature on how we can learn about the causal structure, known as causal inference (15) or causal discovery when we have minimal prior knowledge or assumptions of which parent-child relationships do and do not obtain [43). The literature also contains a number of methods for assessing counterfactual fairness given a partial or complete causal graph 20 52 57]. Again, we do not attempt to resolve debates about appropriate causal inference strategies in the present work. but merely to provide a conceptual tool for those researchers and practitioners who are comfortable staking a claim of some partial or complete causal structure of the context at hand.
To ground our technical results, we briefly sketch three fairness contexts that match those in Figure 2 and will be the basis Corollary 2.2] First, measurement error (6) is a well-known source of fairness issues as developed in the "measurement error models" of Jacobs and Wallach [26]. Suppose that COMPAS is assessing the risk of recidivism, but they do not have perfect knowledge of who has committed crimes because not all crimes committing of a crime, which is one cause of Y, the imperfect labels. Also suppose that the protected class A affects whether the crime is reported and prosecuted, such as through police bias, but does not affect the actual committing of a crime. A also affects X, other data used for the recidivism
reported and prosecuted. Thus, X causes Y, the actual
5

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prediction. This is represented by the causal DAG in Figure 2a which connects counterfactual fairess demographic parity.
Second, protected class can affect whether individuals are selected into the dataset. For example, drugs may be prescrihed the basis of whether the individual's lab work X indicates the presence of disease Y. People with the disease are presumably more likely to have lab work done, and the socioeconomic status A of the person (or of the hospital where their lab work is done) may make them more likely to have lab work done. We represent this with a selection variable S that we condition on by only using the lab work data that is available. This collider induces an association between A and Y. This is represented by the causal DAG in Figure 2b which connects counterfactual fairness to equalized odds.
Third, individuals may be selected into the dataset based on predictors, X, which cause the label Y. For example, loans may be given out with prediction of loan repayment Y based on demographic and financial records X. There may be data only for people who choose to work with a certain bank based financial records and a protected class A. This represented by the causal DAG in Figure 2c which connects counterfactual fairness to calibration.
4 Counterfactual fairness and robust prediction
Characterizations of the trade off between prediction accuracy and fairness treat them as two compet- That view assumes the task is to find an optimal model f*(X) that minimizes risk (i.e., expected loss) in the training distribution X, Y,
ing goals [e.g.. 11 12 21 55
(X) argmin Ep({([(X))))
However, the discrepancy between levels of the protected class in the training data may itself due to biases in the dataset, such as measurement error, selection on label, and selection predictors. As such, the practical interest may not be risk minimization in the training distribution, but out-of- distribution (OOD) generalization from the training distribution to an unbiased target distribution where these effects are not present.
Whether the counterfactually fair empirical risk minimizer also minimizes risk in the target distribution depends on how the distributions differ. Because counterfactually fair predictor does not depend on the protected class, it minimizes risk if the protected class in the target distribution contains no information about the corresponding labels (i.e., if protected class and label are uncorrelated). the protected class and label are correlated in the target distribution, then risk depends whether the counterfactually fair predictor learns all the information about the label contained in the protected class. If then its prediction has reason to vary in the counterfactual scenario. Theorem I motivates counterfactual fairness by stating that, for a distribution with bias due to selection on label and equal marginal label distributions or due to selection on predictors, the counterfactually fair predictor is accuracy-optimal in the unbiased target distribution.
Theorem 1. Let JCF be the set of all counterfactually fair predictors. Let E be a proper scoring rule (e.g. square error, cross entropy loss). Let the counterfuctually fair predictor that minimizes risk on the training distribution X, A ~ be:
(x(X) == argmin Ep(e(f(X), Y)]
fEFO
Then, f also minimizes risk on the target distribution X,Y, with no selection effects, i.e.,
f*( argmin Egl((f(x),
either of the following conditions hold:
1. The association between Y and A is due selection label and the marginal distribution of the label is the same in each distribution, i.e., P(Y)= Q(Y).
2. The association between Y and A is due to selection on predictors.
In the context of measurement error, there are not straightforward conditions for the risk minimization of the counterfactually fair predictor because the training dataset contains Y, observed noisy labels,

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or d-separates and not Y. the true unobserved labels. Thus any risk minimization (including in the training distribution) depends on the causal process that generates Y from Y.
5 Counterfactual fairness and group fairness
Causal structures imply conditional independencies. For example, if the only causal relationships are that X causes Y and Y causes Z. then we know that X 1 Z Y. On directed acyclic graph (DAG), following Pearl and Dechter [38], we say that a variable dependence-separre X and Z if X and Z are connected via an unblocked path (i.e., no unconditioned collider in which two arrows point directly to the same variable) but are no longer connected after removing all arrows that directly connect to Y [22]. So in this example, Y d-separates X and Z, which is equivalent to the conditional independence statement. A selection variable S in a rectangle indicates an observed in the data-generating process, which induces association between its parents and thereby does not block the path as an unconditioned collider would. For For [Theorem 2 Corollary 2. and Corollarry 2.2] make the usual assumption of faithfulness of the causal graph, meaning that the only conditional independencies are those implied by d-separation, rather than any causal effects that happen to be precisely counterbalanced. Specifically, we assume faithfulness between the protected class A, the label Y, and the component of X on which the predictor f(X) based. If the component is not already its own node in the causal graph, then faithfulness would apply if the component were isolated into its own node or nodes.
variable that is conditioned
For a predictor, these implied conditional independencies can be group fairness metries. We can restate conditional independencies containing x as containing I' [(X) because, if f(X) is a counterfactually fair predictor, it only depends on q the component that is not causally affected by A. In Theorem 2
we provide the correspondence between counterfactual fairness and the three most common metrics: demographic parity, equalized odds, and calibration.
Theorem Let the causal structure be a causal DAG with X X.,, Y, and A, such as Figure Assume faithfulness between A, Y, and f(X). Then:
1. Counterfactual fairness is equivalent to demographic parity if and only if there is no unblocked path between X and A.
2. Counterfactual fair fairness is equivalent to equalized odds if and only if all paths between X and A. if any, are either blocked by a variable other than Y or unblocked and contain Y.
3. Counterfactual fairness is equivalent to calibration if and only if all paths between Y and A, if any, are either blocked by a variable other than X X or unblocked and contain Xt.
Similar statements be made for any group fairness metric. For example, the notion of false negative error rate balance, also known as equal opportunity [23), is identical to equalized odds but only considers individuals who have positive label (Y 1). The case for this metric is based on false negatives being a particular moral or legal concern, motivated by principles such as "innocent until proven guilty," " in which false negative represents an innocent person (Y 1) who is found guilty (X) = 0). False negative error rate balance is assessed in the same way as equalized odds but only with observations that have positive true label, which may be consequential if different causal structures are believed to obtain for different groups.
Table shows the ten group fairness metrics presented in Verma and Rubin 48 that can be ex- pressed as conditional independencies. [Theorem 2]specified the correspondence between three of these (demographic parity, equalized odds, and calibration), and we extend to the other seven in Corollary 2.1
We have so far restricted ourselves to a binary classifier f(X) E 10. Here, we denote this as decision fxx = d E = {0, 1} and also consider probabilistic classifiers that produce score s that can take any probability from 0 to 1, i.e., f(r)
E S = [o. 1] P(Y = 1). The metrics of balance for positive class, balance for negative class, and score calibration are defined by Verma and Rubin (48) in terms of score. Verma and Rubin (48) refer to calibration for binary classification (Definition 3) as "conditional use accuracy equality." " To differentiate these, we henceforth refer to the binary classification metric as "binary calibration" and the probabilistic classification metric as "score calibration."
7

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Table Group faimesss metrics from Verma and Rubin [48]
Independence Definition
Name
Probability Definition
Demographic Parity
PP = A P
D A
Conditional
= - y)
D A LLAL-1
Demographic Parity
Equalized Odds
P(D 1 A A=
DLAY Y
False Positive
= = -0=
0)
D D
Error Rate Balance
False Negative Error Rate Balance
P(D = 0 A = 0,Y = -
D DLA PLAY=1
Balance for
E A 0,Y = E|s A 1 =0
$ SLAIY =
Negative Class
Balance for
E A = 0,Y = 1|
= 1-1Y
$ SLAYY=
Positive Class
Conditional Use Accuracy Equality
P(Y = y = = A = 1,D d)
Y P LAA LADD
(i.e., Calibration)
Predictive Parity
P(Y -)=
1,0 1)
Y Y ALAID Y
Score Calibration
PP(Y =
= P(Y = P 1
Y YLASS
Corollary 2.1. Let the causal structure be causal DAG with Xy. X Y, and A, such as in
Figure?
Assume faithfulness between A, Y, and f(X). Then:
1. For a binary classifier, counterfactual fairness is equivalent conditional demographic parity if and only if. when a set of legitimate factors Lis held constant at level there is no unblocked path between X and A.
2. For a binary classifier, counterfactual fairness is equivalent to false positive error rate balance if and only if. for the subset of the population with negative label (i.e., Y = 0), there is no path between X and A, a path blocked by variable other than Y, or an unblocked path that contains Y.
3. For binary classifier, counterfactual fairness is equivalent false negative error rate balance if and only if. for the subset of the population with positive label (i.e., Y 1), there is no path between X and A, a path blocked by variable other than Y, or an unblocked path that contains Y.
4. For a probabilistic classifier, counterfactual fairness is equivalent to balance for negative class if and only if. for the subset of the population with negative label (i.e., Y 0), there no path between X A and A, a path blocked by a variable other than Y, or unblocked path that contains
5. For a probabilistic classifier, counterfactual fairness is equivalent to balance for positive class if and only if. for the subset of the population with positive label (i.e., Y there no path between X and A, a path blocked by a variable other than Y, or an unblocked path that contains Y.
6. For a probabilistic classifier, counterfactual fairness equivalent to predictive parity if and only if. for the subset of the population with positive label (i.e., D = 1), there is path between Y and A, a path blocked by a variable other than X or an unblocked path that contains X
7. For a probabilistic classifier, counterfactual fairness is equivalent to score calibration if and only if there is path between Y and A. path blocked by variable other than X A unblocked path that contains X

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Table 2: Experimental results: Robust prediction
Source Accuracy
Target Accuracy
Target Trained
Naive
FTU
CF
Naive
FTU
CF
Measurement Error
0.8283
0.7956
0.7695
0.8031
0.8114
0.8204
0.8674
Selection on Label
0.8771
0.8686
0.8610
0.8566
0.8652
0.8663
0.8655
Selection on Predictors
0.8659
0.656
0.8658
0.8700
0.8698
0.8699
0.8680
This general specification can be unwieldy, so to illustrate this, provide three real-world examples shown in Figure 2 measurement error, selection on label (i.e., "post-treatment bias" or "selection on outcome" if the label Y is causally affected by X). and selection on predictors. These imply demographic parity, equalized odds, and calibration, respectively.
Corollary 2.2. Assume faithfilness between A, Y. and f(X).
1. Under the graph with measurement error as shown in Figure 2a counterfactual fairness if and only if achieves demographic parity.
a predictor achieves
2. Under the graph with selection on label as shown in Figure 2b counterfactual fairness if and only it achieves equalized odds.
predictor achieves
3. Under the graph with selection on predictors as shown in Figure
2c a predictor achieves
counterfactual fairness if and only if achieves calibration.
Experiments
We conducted brief experiments in semi-synthetic setting to show that counterfactually fair predictor achieves robust prediction and group fairness. We used the Adult income dataset (3) with a simulated protected class A, balanced with P(A = 0) P(A = 1) 0.5. For observations with A = 1. we manipulated the input data to simulate a causal effect of A on X: P(race = Other) = 0.8. With this the target distribution, effect produced with a fixed probability for observations when A = 1: measurement error (P = 0.8), selection label (P 0.5), and selection predictors (P 0.8).
produced three biased datasets that result from each
On each dataset, we trained three predictors: naive predictor trained on A and X, fairness through unawareness (FTU) predictor trained only on X, and counterfactually fair predictor based on an average the naive prediction under the assumption that = and the naive prediction under assumption A O, weighted by the proportion of each group in the target distribution.
Theorem 3. Let X be an input dataset X E X with a binary label Y E y = {0, 1} and protected
class A E {0, 1}. Define a predictor: fnaive argmin [[((((XX
where f is proper scoring rule. Define another predictor:
for = P(A 1)fnaine (X, 1) P(A 0)f (X,0)
If the association between Y and A is purely spurious, then for is counterfactually fair.
For robust prediction, we show that the counterfactually fair (CF) predictor has accuracy the target distribution near the accuracy of a predictor trained directly on the target distribution. For group fairness, we show that the counterfactually fair predictor achieves demographic parity, equalized odds, and calibration, corresponding and Table 3, and code to reproduce these results or produce results with varied inputs (number of datasets sampled, effect of A on X, probabilities of each bias, type of predictor) is available https///athut.cmmjjyyaatiiCCCCuu
the three biased datasets. Results are shown in Table 2

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Table 3: Experimental results: Fairness metrics
Demographic
Equalized
Calibration Difference (CF)
Parity Difference (CF)
Odds Difference (CF)
Mcasurement Error
-0.0005
0.0906
-0.8158
Selection on Label
0.1321
-0.0021
0.2225
Selection on Predictors
0.1428
0.0789
0.0040
7 Discussion
In this paper, we provided new argument for counterfactual fairness- the supposed trade- off between fairness and accuracy [12] can evaporate under plausible conditions when the goal is accuracy in an unbiased target distribution (Theorem I). To address the challenge of trade-offs between different group faimess metrics, such as their mutual incompatibility [30) and the variation costs of certain errors such as false positives and false negatives [19]. we provided a conceptual tool for adjudicating between them using knowledge of the underlying causal context of the social problem (Theorem 2 and Corollary 2.Ip. We illustrated this for the three most common fairness metrics, in which the bias of measurement error implies demographic parity; selection on label implies equalized odds; and selection on predictors implies calibration (Corollary 2.2p, and we showed minimal example by inducing particular biases simulated protected class in the Adult income dataset.
There nonetheless important limitations that we hope can be addressed in future work. First, the counterfactual fairness paradigm still faces significant practical challenges, such as ambiguity and identification, Researchers can use causal discovery strategies developed in fairness context [5]20) 20] to identify the causal structure of biases in real- world datasets and ensure theory like that outlined in this paper can be translated application. Second, key assumption in our paper and related work has been the assumption that associations between Y and A are "purely spurious," term coined by Veitch al. [47] to refer to cases where, if one conditions on A, then the only information about Y in X is in the component of X that is not causally affected by A. This has provided useful conceptual foothold to build theory, but should be possible for future work to move beyond the purely spurious case, such as by articulating distribution shift and deriving observable signatures of counterfactual fairness more complex settings 36 39
Acknowledgments
We thank Bryon Aragam, James Evans, and Sean Richardson for useful discussions.
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A Proofs
Theorem Let JCC be the set of all counterfactually fair predictors. Let l be a proper scoring rule (e.g, square error, cross entropy loss). Let the counterfuctually fair predictor that minimizes risk on the training distribution X, A be:
(*) == argmin Ep(e(/(X),
fEJo
Then, f also minimizes risk the target distribution X, Y. Q with selection effects, i.e.
(*) = argmin Bo(e(((X))Y)
either of the following conditions hold:
1. The association between Y and A is due to selection on label and the marginal distribution of the label is the same in each distribution, i.e., P(Y) Q(Y).
2. The association between Y and A is due to selection on predictors.
Proof. Counterfactual fairness is a case of counterfactual invariance. By Lemma 3.1 in Veitch et al. X --maasurable. Therefore.
47 this implies X is X
argmin Ep(((f(X), argmin Ep(((f(X)),
SE
JEF
Following the same reasoning as Theorem 4.2 in Veitch et al. 47 it is well-known that under squared error or cross entropy loss the risk minimizer is (=xx Er-(Y xx)- Because the target distribution Q has no selection (and no confounding because A is exogenous in the case of counterfactual fairness), the risk minimizer in the target distribution is the same the counterfactually fair risk minimizer the target distribution, i.e., Bo[y |x] = EE[y **) Thus our task is to show [P[YY] x = FY =
Selection on label is shown in Figure 2b Because X does not d-separate Y and A, (X) depends on the marginal distribution of Y so we need an additional assumption that P(Y) Q(Y). We can this with Bayes' theorem to show the equivalence of the conditional distributions,
Q(X Y)Q(Y)
Q(Y xx)
(A.1)
SQ(X* Y)Q(Y)dy
P(X) Y)Q(Y)
(A.2)
J Y)Q(Y)dy
P(X) Y)P(Y)
(A.3)
JP(X) Y)P(Y)dy
= P(Y +
(A.4)
where the first and fourth lines follow from Bayes' theorem, the second line follows from the causal structure (X causes Y), and the third line follows from the assumption that P(Y) = QYY). This equality of distributions implies equality of expectations.
Selection on predictors shown Figure 2c Because X d-separates Y and A, (x) does not depend on the marginal distribution of Y, so we immediately have an equality of conditional distributions, Q(Y x = P(Y x)), and equal distributions have equal expectations, [Y = EY AA)
Theorem 2. Let the causal structure be a causal DAG with X X Y, and A, such as in Figure Assume faithfulness between A, Y, and f(X). Then:
1. Counterfactual fairness is equivalent to demographic parity if and only if there is no unblocked path between X and A.
2. Counterfactual fairness is equivalent to equalized odds if and only if all paths between X and A. if any, either blocked by a variable other than Y or unblocked and contain Y.
14

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A
Y
X
Y
x
(a) Example an unblocked path between
XA (b) Example of blacked path between A and XA
Figure 3: Examples of an unblocked path and blocked path in causal DAG.
3. Counterfactual fairness is equivalent to calibration if and only if all paths between Y and A. if any, are either blocked by variable other than X or unblocked and contain X
Proof. For a predictor f(X) to be counterfactually fair, must only depend on X
1. By Definition 1 demographic parity is achieved if and only if X 1 A. In DAG, variables are dependent and only if there is an unblocked path between them (i.e., no unconditioned collider in which two arrows point directly to the same variable). For example, Figure 3a shows an unblocked path between A and X
2. By Definition 2| equalized odds are achieved if and only if X 1 A I... If there is no path between X and A, then X and A are independent under any conditions. If there is blocked path between X and A, and it has a block that is not Y, then X and A remain independent because blocked path induces no dependence. If there is an unblocked path between X and A that contains Y, then Y d-separates X A and A, so X and A remain independent when conditioning on Y. On the other hand, if there a blocked path between X and A and its only block is Y, then conditioning on the block induces dependence. If there is an unblocked path that does not contain Y, then X1 and A are dependent.
3. With Definition 3
we can apply analogous reasoning to the case of equalized odds. Cali- bration is achieved if and only if Y 1 A X If there no path between Y and A, then Y and A are independent under any conditions. If there is blocked path between Y and A, and has a block not A. then and remain independent because blocked path induces no dependence. If there is an unblocked path between and A that contains X x then X I d-separates Y and A, Y and A remain independent when conditioning X q On the other hand, if there is blocked path between Y and A and its only block is X then conditioning on the block induces dependence. If there is an unblocked path that does not contain X_ then Y and A are dependent.
Corollary 2.1. Let the causal structure be causal DAG with X X Y, and A, such in Figure Assume faithfulness between A, Y, and f(X). Then:
1. For a binary classifier, counterfactual fairness is equivalent conditional demographic parity if and only if. when a set of legitimate factors L is held constant at level I, there is no unblocked path between X and A.
2. For a binary classifier. counterfactual fairness is equivalent to false positive error rate balance if and only if. for the subset of the population with negative label (i.e.. Y = 0), there is no path between X X and A, a path blocked by variable other than Y, or an unblocked path that contains Y.
3. For a binary classifier, counterfactual fairness is equivalent to false negative error rate balance if and only for the subset of the population with positive label (i.e., Y = =1, there is no path between X and A, a path blocked by variable other than Y, or an unblocked path that contains Y."
4. For a probabilistic classifier, counterfactual fairness is equivalent balance for negative class if and only if. for the subset of the population with negative label (i.e., Y 0), there is no path between X, and A, a path blocked by a variable other than Y, or an unblocked path that contains Y.
15

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5. For a probabilistic classifier, counterfactual fairness is equivalent to balance for positive class if and only if. for the subset of the population with positive label (i.e., Y 1). there path between X and A, a path blocked by a variable other than Y, or unblocked path that contains Y.
6. For a probabilistic classifier, counterfactual fairness equivalent to predictive parity if and only if. for the subset of the population with positive label (i.e.. D= 1). there is no path between Y and A, a path blocked by a variable other than A or an unblocked path that contains X
7. For a probabilistic classifier, counterfactual fairness equivalent score calibration if and only if there is no path between Y and A, path blocked by variable other than X A' or an unblocked path that contains X
Proof. Each of these seven metries can be stated as conditional independence statement, as shown in Table I] and each of the seven graphical tests of those statements can be derived from one of the three graphical tests Theorem 2) Note that the graphical test for binary classifier the same as that for the corresponding probabilistic classifiers because the causal graph does not change when f(xx) changes from binary-valued (i.e., f(X) E {0, 1}) function a probability-valued function (i.e., f(X) E [0, 1).
From demographic parity:
Conditional demographic parity is equivalent to demographic parity when some set of legitimate factors L is held constant at some value 1.
From equalized odds:
2. False positive error rate balance is equivalent to equalized odds when considering only population with negative label (i.e., Y 0).
3. False negative етror rate balance is equivalent to equalized odds when considering only the population with positive label (i.e., Y 1).
4. Balance for negative class is equivalent equalized odds for probabilistic classifiers when considering only the population with negative label (i.e., Y = 0).
5. Balance for positive class is equivalent to equalized odds for probabilistic classifiers when considering only the population with negative label (i.e., Y =).
From binary calibration:
6. Predictive parity is equivalent to binary calibration when considering only the population with positive label (i.e., D 1).
7. Score calibration is equivalent to binary calibration for probabilistic classifiers.
Corollary 2.2. Assume faithfulness between A, Y, and f(X).
1. Under the graph with measurement error as shown in Figure 2a counterfactual fairness if and only it achieves demographic parity.
a predictor achieves
2. Under the graph with selection on label as shown in Figure 2b counterfactual fairness if and only if it achieves equalized odds.
predictor achieves
3. Under the graph with selection on predictors as shown in Figure counterfactual fairness and only if achieves calibration.
2c predictor achieves
Proof. By Theorem 2
16

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1. Observe in Figure 2althat the only path between X A and A is blocked by Y. so counterfactual faimess implies demographic parity. Because the only block in that path fairness does not imply equalized odds. And the only path between Y and A (a parent-child relationship) unblocked and does not contain X so counterfactual fairness does not imply calibration.
Y, counterfactual
2. Observe in Figure 2b that the only path between X A and A is unblocked (because S is necessarily included in the predictive model), so counterfactual fairness does not imply demographic parity. Because that path contains Y, counterfactual fairness implies equalized odds. And the only path between and unblocked (because necessarily included in the predictive model) and does not contain X so counterfactual fairness does not imply calibration.
Observe in Figure 2c that the only path between X and is unblocked (because S necessarily included in the predictive model), so counterfactual fairness does not imply demographic parity. Because that path does not contain Y, counterfactual fairness does not imply equalized odds. And the only path between Y and A is unblocked (because S is necessarily included the predictive model) and contains X so counterfactual fairness implies calibration.
Theorem 3. Let X be input dataset X X with binary label Y E {0, 1) and protected class A E {0, 1}. Define a predictor:
fnaive argmin ((C
where f is proper scoring rule. Define another predictor:
(X,1) for = (AA 1) (X, 1) + P(A O) (X,
the association between Y and A is purely spurious, then fer counterfactually fair.
Proof. Notice that fcr does not depend on A directly because the realization of A is not in the definition. To show that fcr also does not depend on indirectly (i.c., through X), consider that purely spurious association means that Y T X X A. Therefore, the naive predictor:
fnaive (octee(X,, A) = P(Y = 1
PP = 1)
Because X is the component of X that is not causally affected by A, there is term in for that depends on A, which means for is counterfactually fair.
17


Reviews:
[{"review_id": "dEo63e6Zu0", "reviewer": "Reviewer_WTYn", "summary": "This paper provides a new way to analyze the fairness-accuracy tradeoff in many fairness definition through the lens of counterfactual fairness. The authors prove that, under certain conditions, the counterfactually fair predictor is optimal when the target is OOD generalization. The theoretical analyses build the connection between counterfactual fairness and 10 well-known correlational fairness.\n", "questions": "Q1. I agree that assumptions about the purely spurious relationship between $Y$ and $A$ helps with theorem 1. But for the race and zip code example in lines 203 -- 205, can we simply say the bias is due to selection effect? It is obvious that population from different demographic groups are distributed unevenly in a community, and the selection effect may further amplify the bias. Can we simply characterize it as due to selection effect only? \n\nQ2. Is it possible to elaborate more on the assumption about decomposing $X$ into $X_{\\perp A}$ and $X_{\\perp Y}$? Is it a linear decomposition of $X$? It is a pretty strong assumption to decompose $X$ and I am not sure whether it is reasonable or not.\n\nQ3. What is the relationship between the proposed counterfacutal fairness-based analysis to impossibility theorem? The authors use impossibility theorem as one motivation for studying counterfactual fairness, but no more discussion about how these analyses are related to impossibility theorem is included.\n\nQ4. Where is the evaluation on simulated data? The authors mention it as a contribution but I could not find it. If I miss it, it would be good to provide pointers.\n\nQ5. How does the proposed analyses help **select** the algorithmic fairness metrics? Some discussions (or maybe a case study) about how to use the proposed analyses to select fairness metrics would help.", "limitations": "L1. Many theoretical analysis in the paper assume the decompostion of $X$ into two parts. But it is unclear whether this assumption is reasonable.\n\nL2. It might be good to draw figures for some concepts for better clarity (e.g., illustrative figures for unblocked path and the correspondence between counterfactual fairness and correlational fairness). And it would be good to recall Figure 2 when discussing related technical details (e.g., theorem 2).\n\nL3. The authors mention that the theorems hold on simulated data, but no evaluation is found.\n\nL4. It would be great to better connect the content with the title `select algorithmic fairness metrics'.\n\nL5. Please carefully check typos in the paper, e.g., `if they had if they were classified negatively' $\\rightarrow$ `if they were classfied negatively'.\n\n================== Post-rebuttal ==================\nThe authors have addressed my concern so I am raising my score.\n", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "S1. Interesting way to connect counterfactual fairness, out-of-distribution generalization and other fairness definitions.\n\nS2. Technically sound proof on the theorems and corollaries.\n\nS3. Writing quality is good.", "weaknesses": "W1. Some statements/assumptions need more comprehensive illustration.\n\nW2. Interesting theoretical analyses on counterfactual fairness\n\nW3. The authors use impossibility theorem as one motivation, but more discussion about the technical analyses to impossibility theorem would better position this paper.\n\nW4. Title is a bit misleading to me.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your time and useful suggestions, which significantly improve the manuscript.\n\n> W1. Some statements/assumptions need more comprehensive illustration.\n\nWe have added the more comprehensive illustrations in response to your comments below.\n\n> W3/Q3. The authors use impossibility theorem as one motivation, but more discussion about the technical analyses to impossibility theorem would better position this paper…\n\nWe only intended  the impossibility theorems to illustrate the need to make context-specific fairness decisions in practice because the metrics cannot all be satisfied at once, except in the unrealistic cases of perfect prediction, random prediction, or equal features across groups [1,2]. The technical counterfactual fairness analysis relates to this impossibility by providing a tool to select between the observational metrics based on the situation’s causal context (i.e., the data-generating process). We state the criteria that the causal context has to meet in Theorem 2 and Corollary 2.2. In response to the reviewer’s helpful suggestion, we have added a brief explanation of this relationship to the Discussion. We have also reduced the emphasis on impossibility theorems as a motivation in the Abstract and Introduction because there are many other reasons why we need to select between fairness metrics and the impossibility results may not be as relevant in practice as it may seem [e.g., 3].\n\n[1] Chouldechova. 2017. “Fair Prediction with Disparate Impact: A Study of Bias in Recidivism Prediction Instruments.” Big Data. doi:10.1089/big.2016.0047.\n\n[2] Kleinberg, Mullainathan, and Raghavan. 2016. “Inherent Trade-Offs in the Fair Determination of Risk Scores.” ITCS. doi:10.48550/arXiv.1609.05807.\n\n[3] Andrew, Bynum, Drushchak, Zakharchenko, Rosenblatt, and Stoyanovich. 2023. “The Possibility of Fairness: Revisiting the Impossibility Theorem in Practice.” FAccT. doi:10.1145/3593013.3594007.\n\n> W4/L4. Title is a bit misleading to me … better connect the content with the title `select algorithmic fairness metrics'.\n\nWe plan to retitle the paper “Causal Context Connects Counterfactual Fairness to Group Fairness and Robust Prediction.”\n\n> Q1. I agree that assumptions about the purely spurious relationship between Y and A helps with theorem 1. But for the race and zip code example in lines 203 -- 205, can we simply say the bias is due to selection effect? It is obvious that population from different demographic groups are distributed unevenly in a community, and the selection effect may further amplify the bias. Can we simply characterize it as due to selection effect only?\n\nWe appreciate you pointing out that correlations between race and recidivism might be due to other factors in addition to the selection effect. We have softened the language here and clarified our statement that “other factors” can have effects because we agree that, in the real world, it is probably not only due to a selection effect.\n\n> Q2/L1. Is it possible to elaborate more on the assumption about decomposing X into X \\perp A and X \\perp Y? Is it a linear decomposition of X?...\n\nWe have added more detail when we first mention decomposition. We do not mean a necessarily linear composition. The decomposition is quite general, and does not specify a particular functional form. E.g., $X^\\perp_A$ may be some arbitrarily complicated nonlinear function $g(X)$, so long as $g(X(A=0)) = g(X(A=1))$ a.s. A key advantage of the view in this paper is that we do not need to know what this function is explicitly. The decomposition into two (abstract) parts is the ‘purely spurious’ assumption—see the global response for a discussion. We have added text to clarify this in the paper.\n\n> Q4/L4. Where is the evaluation on simulated data? The authors mention it as a contribution but I could not find it…\n\nWe apologize for the erroneous statement left over from a draft that had simulations. We have in fact now added semi-synthetic experiments based on the Adult income dataset, detailed in the global response.\n\n> Q5. How does the proposed analyses help {\\bf select} the algorithmic fairness metrics? Some discussions (or maybe a case study) about how to use the proposed analyses to select fairness metrics would help.\n\nIn short: the analyst uses their understanding of the real-world situation to posit a causal graph, then selects the fairness metric implied by that graph. We give examples in the paper of measurement error in arrest data (e.g., COMPAS), selection on outcome with people of younger age or privileged backgrounds being more likely to be included in the dataset, and selection on predictors with loan repayment data also not being a balanced mix of all protected classes. We discuss real-world application more in the global response.\n\n> L2. It might be good to draw figures for some concepts … recall Figure 2 when discussing related technical details (e.g., theorem 2).\n\nWe have added a brief illustration of step-by-step reasoning to connect counterfactual fairness to observational fairness. (e.g., an example of an unblocked path). We have also added an explicit mention of Figure 2 to the statement of Theorem 2.\n\n> L5. Please carefully check typos in the paper, e.g., ```if they had if they were classified negatively' $\\rightarrow$``` if they were classfied negatively'.\n\nWe have fixed this and gone through the paper to check for other typos."}, {"review_id": "EH6MB2p7uz", "reviewer": "Reviewer_U9Dm", "summary": "The paper explores a connection between counterfactual fairness and correlation-based fairness metrics, exhibiting causal graph settings in which counterfactual fairness implies specific correlational metrics are satisfied. The paper further connects this mapping to the accuracy-fairness tradeoff.", "questions": "How can the connections offered in this paper be oeprationalized for real-world cases? Many applications are mentioned, but the value of the connections is only loosely coupled to these.", "limitations": "There is some defensive posturing in the conclusion about the lack of real-world connection, which is probably a limitation to address by improving the connections rather than by disclaiming the possibility of doing this (since making those connections is the ostensible value of the paper).", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "I think the mapping between causal/counterfactual notions of fairness and metric-based fairness is useful. The paper does a good job of integrating several literatures into a coherent frame. The specific causal scenarios that relate counterfactual fairness and correlational fairness are useful, although I'm not well equipped enough in the structural causal modeling literature to understand the significance of this as a contribution.", "weaknesses": "I think the paper itself foresees my largest concern: it's not clear what settings these results apply in. While I'm not suggesting, as the conclusion tries to defend against, a full empirical validation of the theory, it would be nice if there was at least some attempt at mapping the causal graphs in Figure 2 to specific scenarios where the relationship being offered is relevant and an explanation of the relevance in that scenario. This exists, but way down in Section 5. Help your reader by making these connections right away.\n\nThe literature on fairness (the human value) as operationalized by fairness (statistical metrics) strongly suggests that the problem with choosing metrics is less a fairness/accuracy tradeoff or impossibility of satisfying multiple metrics, but the need to contextualize the model well in the application, understanding the cost and meaning of things like errors and the value of their avoidance. It's not obvious that connections to counterfactual fairness help with this problem, and if they don't, that's a limitation worth articulating.\n\nThe paper often cites into the literature in a way that only thinly and tenuously represents the large issues at play in the topic of the paper. I'd like to see more alignment between supporting sources and the content of the paper. For example, the citation of Jacobs & Wallach is offered as a problem with the difficulty of measurement error, but the paper is mostly about how to establish validity of models through explicit measurement modeling, a thing not considered by the methods in this paper at all.\n\nNit: the paper consistently uses the word \"persecuted\" instead of \"prosecuted\" (e.g., at 53, 264, and 266, and maybe elsewhere?)\nNit: the symbol l in Theorem 1 should probably be $\\ell$.\nNit: the discussion of gender at 164 is highly oversimplified and is probably actually a description of biological sex. Later text clarifies this, but the claims are a little vague as currently stated.\nNit: \"obtain\" at 178 is probably the wrong word?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your support and your useful comments.\n\n> I think the paper itself foresees my largest concern: it's not clear what settings these results apply in. … This exists, but way down in Section 5. Help your reader by making these connections right away.\n\nWe agree that the mapping of causal graphs in Figure 2 to specific scenarios is important, and we have moved the example description scenarios to appear sooner in the paper. \n\nWe have also added an experimental demonstration illustrating the results (see the global response), which we think further clarifies the practical scenarios corresponding to the causal graphs.\n\n> The literature on fairness (the human value) as operationalized by fairness (statistical metrics) strongly suggests that the problem with choosing metrics is less a fairness/accuracy tradeoff or impossibility of satisfying multiple metrics, but the need to contextualize the model well in the application, understanding the cost and meaning of things like errors and the value of their avoidance. It's not obvious that connections to counterfactual fairness help with this problem, and if they don't, that's a limitation worth articulating.\n\nThank you for pointing out the importance of addressing the cost and meaning of errors that vary across contexts (e.g., false negatives versus false positives), and we have added that explicitly in the motivation. We believe notions of counterfactuals can potentially help with this problem. For example, the severity of harm from an error can be in part estimated by comparing it to the harm or benefit one would receive in the counterfactual scenario. And if one believes counterfactually unfair practices (i.e., those prohibited by the “but-for” test in U.S. law) lead to more harmful errors, then one might be able to use the correspondence we draw to better achieve that. For example, the types of errors made by a predictor that only achieves demographic parity will tend to be different than those of a predictor that only achieves equalized odds.\n\nWe would push back a little on the problem not being a fairness/accuracy trade-off or the mutual impossibility. In our experience, the literature still views both of these as important challenges [e.g., 1 on the trade-off and 2 on impossibility, just as examples], and we see our work as contributing to the toolkit of methods to address them [e.g., 3 on improving fairness on multiple metrics with “minimal model performance reduction”]. We think that contextual factors, such as the causal graphs that are the focus of this work and other approaches to counterfactual fairness, are relevant for addressing all three of the issues: fairness/accuracy trade-off, the impossibility theorems, and the cost of errors and benefit of correct prediction/classification. We appreciate the reviewer helping us clarify this point, and we have added a brief discussion of this to the paper.\n\n[1] Ge, Zhao, Yu, Paul, Hu, Hsieh, and Zhang. 2022. “Toward Pareto Efficient Fairness-Utility Trade-off in Recommendation through Reinforcement Learning.” WSDM. doi:10.1145/3488560.3498487.\n\n[2] Hsu, Mazumder, Nandy, and Basu. 2022. “Pushing the Limits of Fairness Impossibility: Who’s the Fairest of Them All?” NeurIPS. doi:10.48550/arXiv.2208.12606.\n\n[3] Bell, Bynum, Drushchak, Zakharchenko, Rosenblatt, and Stoyanovich. 2023. “The Possibility of Fairness: Revisiting the Impossibility Theorem in Practice.” FAccT. doi:10.1145/3593013.3594007.\n\n> The paper often cites into the literature in a way that only thinly and tenuously represents the large issues at play … explicit measurement modeling, a thing not considered by the methods in this paper at all.\n\nWe see that J&W are focused on a particular approach that we do not apply or respond to. As far as we know, they have the most rigorous development of “measurement error models” in the fairness context, which is why we cited them. We have edited line 262 to specify that we are referring to their development of such models, and we have also provided a more general citation for measurement error [1]. We also reviewed the other citations in the paper to ensure that they are aligned with our text.\n\n[1] Bland and Altman. 1996. “Statistics Notes: Measurement Error.” BMJ. doi:10.1136/bmj.312.7047.1654.\n\n> Nit: the paper consistently uses the word \"persecuted\" instead of \"prosecuted\" (e.g., at 53, 264, and 266, and maybe elsewhere?) Nit: the symbol l in Theorem 1 should probably be $\\ell$. \n\nFixed.\n\n> Nit: the discussion of gender at 164 is highly oversimplified and is probably actually a description of biological sex. Later text clarifies this, but the claims are a little vague as currently stated. \n\nWe could move the later text to the beginning of the discussion and reword for clarity, but with the existing discussion of race, we have just cut the gamete randomization example to make room for suggested additions.\n\n> How can the connections offered in this paper be oeprationalized for real-world cases? Many applications are mentioned, but the value of the connections is only loosely coupled to these… /// There is some defensive posturing in the conclusion about the lack of real-world connection…\n\nWe have provided a partial connection to real-world cases through:\n1. clarification of how the connections help one to measure (or enforce) counterfactual fairness using observational data,\n2. elaboration of the three example problems, and\n3. an experiment demonstration of the main results (including a simple method for learning a counterfactually fair predictor in the purely spurious setting; see the global response).\n\nWe have also expanded the discussion of what “purely spurious” means in practice and clarified that this assumption is a limitation of the present paper (though not a fundamental one—we think it’s a great direction for future work to extend the ideas here to circumvent this!). We also discuss this in more detail in the global response."}, {"review_id": "hLixZ3ulZU", "reviewer": "Reviewer_7eza", "summary": "This work investigated two important coutnterfactual fairness problems: (1) When will the counterfactually fair predictor also be the risk minimizer on the target distribution? (2) How do the classic statistical fairness metrics correspond to counterfactual fairness? In response to these questions, this work contextualized the problem in three causal DAG where selection bias may exist. To respond the first question, Theorem 1 shows under certain conditions that the counterfactual fair predictor is in fact accuracy-optimal. To respond the second question, Theorem 2 and Corollary 2.2 characterized the counterfactual fairness using the knowledge of the underlying causal context.  The success in responding the two questions will motivate a pipeline to achieve counterfactual fairness without performance degradation.\n\n", "questions": "1. In Figure 2(b), $X^\\perp_A$ is the effect of outcome $Y$. What is the implication of such causal structure in the real world?\n2. In Figure 2(c), the selection variable $S$ is the causal effect of protected class $A$ and observed outcome $Y$. What if the selection $S$ is modeled as the cause of the outcome $Y$? Do the statements still hold?", "limitations": " The authors did not address the limitations of how the counterfactual fairness can be evaluated in the real world.", "rating": 6, "confidence": 2, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "This work bridges the gap between counterfactual fairness and statistical fairness metrics in the presence of selection biases. The studied problem is novel, and the theoretical results are of high quality. The motivation and the presentation of the whole paper is clear, and can provide high-level insights for fairness researchers and practitioners.", "weaknesses": "I did not fully check the proof details but only read the sketch. It looks like the statements heavily depend on the three causal DAGs in Figure 2. For the theoretical results, the assumptions for each statement can be made more clearly.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for the support, corrections, and provocative questions.\n\n> I did not fully check the proof details but only read the sketch. It looks like the statements heavily depend on the three causal DAGs in Figure 2. For the theoretical results, the assumptions for each statement can be made more clearly.\n\nCorollary 2.1 is exclusively about those three causal DAGs, but Theorem 1 and Theorem 2 neither reference nor depend on them. You make a good point that the scope of Corollary 2.1 is not very clear in the statement itself because Figure 2 is only mentioned on Line 245 in the preceding paragraph, so we have added a reference to Figure 2 in the statements to clarify that these statements refer to Figure 2a, Figure 2b, and Figure 2c, respectively.\n\n> Q1. In Figure 2(b), X \\perp A is the effect of outcome Y. What is the implication of such causal structure in the real world?\n\nThis causal structure is referred to as “anti-causal” and is the predominant setting in many machine learning applications. For example, in health care data, we’re often interested in assessing whether or not someone has a disease based on measurements of effects of that disease—e.g., $X$ may be a chest X-ray, and $Y$ some lung disease. Here, $Y$ causes $X$. (Ultimately, the true labels used for the training data may be produced by more expensive follow-up measurements or waiting for the disease to progress.)  \n\n> Q2. In Figure 2(c), the selection variable S is the causal effect of protected class A and observed outcome Y. What if the selection S is modeled as the cause of the outcome Y? Do the statements still hold?\n\nWe assume the reviewer means “and X” rather than “and observed outcome Y” because Figure 2(c) shows $S$ being affected by $X$, not $Y$. If so, note that $S$ is selection into the dataset. In our example, it is whether the loan repayment predictors $X$ and whether the person repaid ($Y$). Because the dataset contains historical data, it doesn’t seem to make logical sense in the real world to say that selection causes loan repayment.\n\nHowever, ignoring the real-world meaning of an effect of $S$ on $Y$, the mathematical answer is that if $S$ caused $Y$ in Figure 2(c), then counterfactual fairness would no longer imply calibration. This is because the new unblocked path between $A$ and $Y$ that does not go through $X_A^\\perp$ makes the graph (i.e., the context) fail the test for calibration in Theorem 2. Additionally, in this case, $A$ would have a true causal influence on $Y$ (i.e., not purely spurious), and therefore it would be unclear whether a counterfactual notion of fairness is appropriate.\n\n> The authors did not address the limitations of how the counterfactual fairness can be evaluated in the real world.\n\nWe describe a number of limitations of how counterfactual fairness can be evaluated in the real world, as detailed in Lines 150–182, beginning, “Counterfactual fairness is widely compelling, but there are still limitations.” We focus on two broad categories of limitation: the ambiguity of counterfactuals and the identification of causal structure. \n\nTo make this more prominent, we have edited the discussion section to bring up these limitations. We would also note that some other revisions have also foregrounded the limitations of generalizability and application (e.g., the “purely spurious” assumption), and we have added an experiment with the Adult income dataset, in which we simulated a new protected class and induced the three causal contexts in Corollary 2.1 and showed that a counterfactually fair predictor achieves the corresponding metric and achieves the best performance in the unbiased target distribution (Theorem 1)."}, {"review_id": "5QO7n21k9q", "reviewer": "Reviewer_buKj", "summary": "The authors show that under various causal graphs involving protected attributes, features and labels, that a counterfactually fair classifier achieves a specific association-based fairness metric. From this, they suggest a pipeline to detect counterfactual fairness. ", "questions": "Can you try to relate the evaporation of the tradeoff that you show to the argument made by Dutta et al. (reference given in the Weaknesses section of the review) and point out similarities and differences. It would also be good to relate the decomposition of X on line 268 to Dutta's other work https://doi.org/10.1609/aaai.v34i04.5794. ", "limitations": "It would be good if the authors could be more explicit on how they obtain/discover the causal graph because it is quite important. That limitation hasn't been discussed enough. As part of that discussion, they may wish to mention causal fairness methods that do not require the entire causal graph a priori and are able to get pieces of it through a group-testing based approach which require a sublinear number of conditional independence tests. ", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The authors do a really good job making the point that counterfactual fairness, which is what is desired in a lot of theory, can be evaluated using more standard associational fairness metrics when assuming (or discovering) particular causal graphs. \n\nThe math seems to be correct.\n\nThe findings are useful to push causal fairness further into practice. ", "weaknesses": "The organization of the paper can be improved. For example, a lot of the beginning of Section 4 would make more sense in the introduction. Figure 2 and its associated introductory text would make more sense in Section 5. \n\nI don't think the impossibility theorems need to be emphasized so much. They're cute, but don't really give us anything practical, which this paper tries to do.\n\nFigure 2: please put the corresponding correlational fairness metric in the main figure rather than only mentioning them in the caption.\n\nLine 48: I don't think Ref [3] does data augmentation. However, this reference does: https://doi.org/10.1145/3375627.3375865. \n\nFigure 1 mentions data generation and regularization, but they don't come up in the text. I think bringing up examples of each would be useful to the reader, e.g. https://proceedings.mlr.press/v161/ahuja21a.html and https://arxiv.org/abs/2002.10774.\n\nIt would be useful to discuss other theoretical work on the fairness-accuracy tradeoff, especially papers that relate this discussion to the presence or absence of different kinds of bias, e.g. https://proceedings.mlr.press/v119/dutta20a.html. \n\nI understand the difficulty of empirical validation when dealing with counterfactuals, but the paper would be a lot stronger if the pipeline of Figure 1 were carried out to show the steps tangibly. ", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for the support and feedback on framing and content, which we believe has improved the paper.\n\n> The organization of the paper can be improved. For example, a lot of the beginning of Section 4 would make more sense in the introduction. Figure 2 and its associated introductory text would make more sense in Section 5.\n\nPer your suggestion, we have moved parts of Section 4 into the introduction. For the second move, we have moved discussion of the three scenarios (5.1, 5.2, 5.3) to the introduction in response to Reviewer U9Dm’s suggestion to, “Help your reader by making these connections right away,” which we think addresses your concern. We would also prefer Figure 2 to be early in the paper because it is so illustrative, if possible. We have also done some other reorganization as a result of other comments.\n\n> I don't think the impossibility theorems need to be emphasized so much. They're cute, but don't really give us anything practical, which this paper tries to do.\n\nThe impossibility theorems motivate the need to make trade-offs between fairness metrics, but we probably did overemphasize them so we have now reduced the emphasis. We have also added a qualification here with reference to a 2023 FAccT paper that critiques their practical relevance [1] alongside the already-cited work that suggests a method to improve fairness across the different metrics “with minimal model performance reduction” [2]. We think this improves the motivation and contextualization of our results.\n\n[1] Bell, Bynum, Drushchak, Zakharchenko, Rosenblatt, and Stoyanovich. 2023. “The Possibility of Fairness: Revisiting the Impossibility Theorem in Practice.” FAccT. doi:10.1145/3593013.3594007.\n\n[2] Hsu, Mazumder, Nandy, and Basu. 2022. “Pushing the Limits of Fairness Impossibility: Who’s the Fairest of Them All?” NeurIPS. doi:10.48550/2208.12606.\n\n> Figure 2: please put the corresponding correlational fairness metric in the main figure rather than only mentioning them in the caption.\n\nDone.\n\n> Line 48: I don't think Ref [3] does data augmentation. However, this reference does: [link].\n\nThank you for noticing this mistake. We have replaced it with your citation, Sharma et al. (2020).\n\n> Figure 1 mentions data generation and regularization, but they don't come up in the text. I think bringing up examples of each would be useful to the reader, e.g. [link] and [link].\n\nWe have added brief mentions and citations to both Ahuja et al. (2021) and Di Stefano et al. (2020).\n\n> It would be useful to discuss other theoretical work on the fairness-accuracy tradeoff, especially papers that relate this discussion to the presence or absence of different kinds of bias, e.g. [link]. /// Can you try to relate the evaporation of the tradeoff that you show to the argument made by Dutta et al. (reference given in the Weaknesses section of the review) and point out similarities and differences.\n\nThank you for the pointer; we’ve added a reference in the paper. The analysis of mismatched distributions by Dutta and colleagues provides an interesting complement to our own critique of the purported fairness-accuracy tradeoff. They base it in the different Chernoff information between groups, and we base ours on the performance of a counterfactually fair predictor in an unbiased target distribution. The two views are complementary is that one is grounded in information theory, the other in causal structure. We think that the results do not have a crisp mathematical connection in general. \n\n> I understand the difficulty of empirical validation when dealing with counterfactuals, but the paper would be a lot stronger if the pipeline of Figure 1 were carried out to show the steps tangibly.\n\nWe have added an experiment with the Adult dataset illustrating the main claims of the paper. This stops short of illustrating all the steps (there’s limited space), but we think it helps clarify the empirical implications. \n\n> It would also be good to relate the decomposition of X on line 268 to Dutta's other work [link].\n\nDutta and colleagues provide an information-theoretic decomposition that usefully separates discrimination into critical features that one wants to preserve and non-critical features that one is okay with being removed. Our decomposition is instead based in a causal framework, which separates the training data $X$ into the component not causally affected by $A$ and the component not causally affected by $Y$, respectively. We have added a brief discussion and citation to Dutta’s approach, which could be explored in future work.\n\n> It would be good if the authors could be more explicit on how they obtain/discover the causal graph … a group-testing based approach which require a sublinear number of conditional independence tests.\n\nOur technical results apply regardless of how the causal graph is obtained/discovered. Generally, we are envisioning that the analyst will use their understanding of the real-world situation under consideration to inform the causal graph they assume. We agree the need to (partially) articulate the causal graph is a limitation. It is a good point that the causal graph itself can be (partially) learned from data, and we have added a note that this is an interesting direction for future work, as well as a citation to Galhotra et al. (2022) [1], which develops a group-testing approach with sublinear conditional independence tests as you suggest.\n\n[1] Galhotra, Shanmugam, Sattigeri, and Varshney. 2022. “Causal Feature Selection for Algorithmic Fairness.” SIGMOD/PODS. doi:10.1145/3514221.3517909."}, {"review_id": "UaJVgfrr5f", "reviewer": "Reviewer_c2Rt", "summary": "This work aims to provide guidance for contextually-informed algorithmic fairness criteria selection. The work motivates counterfactual fairness from its legal and ethical foundations. The primary contribution of the work is to detail a set of cases (data generating processes and selection mechanisms) where counterfactually fair predictors are accuracy-optimal in the (unselected) target distribution and consistent with specific correlational fairness criteria in the observed data.", "questions": "*  To what extent should the statements about the observable implications of counterfactual fairness be considered if and only if statements versus unidirectional statements? For instance, under Corollary 2.1.1 does demographic parity imply counterfactual fairness under the causal measurement error graph?\n* Could you please add additional detail to prove the result regarding predictive parity? I generally follow the claim that binary calibration for binary classifiers is analogous to score calibration for risk scores, but it is not clear how conditioning on Y=1 yields predictive parity (which is confusingly referred to as D=1 in the supplement). It is also unclear whether the argument assumes that score calibration implies predictive parity. In general, this is not true (see Chouldechova 2017 Arxiv version, discussion immediately following definition 2.2 [1]).\n\n1. Chouldechova, Alexandra. \"Fair prediction with disparate impact: A study of bias in recidivism prediction instruments.\" arXiv preprint arXiv:1703.00056 (2017).\n\nMinor suggestions:\n*  Introduce the notion that a counterfactually fair predictor depends only on X_{A}^{\\perp} in the main text rather than in the supplementary material\n*  Replace “correlational” with “observational”\n*  Replace “persecuted” with “prosecuted”\n", "limitations": "It would be beneficial for this work to expand upon the potential limits of the generalizability of its claims, especially due to the assumption of the purely spurious setting. Relatedly, it is not clear how well the claims are applicable outside of the set of graphs and selection mechanisms considered in this work. ", "rating": 6, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "*  The work is generally well-written and clear.\n* The arguments and key results of this work provide a new perspective on cases where counterfactual fairness is consistent with predicting outcomes under particular selection mechanisms and data generating processes. While the results regarding the mapping between counterfactual fairness and observational fairness criteria largely follow from prior work on counterfactual invariance, this work does expand upon the prior work meaningfully.", "weaknesses": "Major:\n* There are no experiments with either simulated or real-world data. At a minimum, it would be beneficial to include simulated experiments that verify the basic claims regarding the optimality of the counterfactually-fair predictors and the relationships between counterfactual fairness and the observational fairness criteria under different data generating processes. \n* The restriction of the scope of the work to the purely spurious setting (line 201) limits the generalizability of the claims and this is largely not acknowledged. The work makes the claim that the purely spurious setting where association between group attributes and outcomes are due to selection is reasonable to cover the space of problems where fairness is of interest. I would argue that this is a serious limitation of the work because of the aim to provide guidance that broadly applies to the space of settings where it is of interest to evaluate fairness and bias considerations.\n  *  For an example where the spurious setting may not hold, consider that in medical applications, structural racism and social determinants of health contribute to differences in observed health status across racial groups in addition to the differences in access to care that may be considered as selection effects. This can induce a difference in the conditional Y | X across subgroups (effectively an association between A and Y not mediated by X) that arises because race serves as a proxy for exposure to structural racism and social determinants. To the best of my understanding, this setting does not map cleanly on to the type of selection graphs considered in this work.\n\nMinor:\n* There is inconsistency in which variable is used to the ground truth outcome versus the noisy outcome with measurement error. In lines 112-113, Y-tilde is considered the noisy label and Y the ground truth. Elsewhere in the paper, Y-tilde is considered the ground truth.\n", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for the close read. We appreciate your time, and we believe that we have addressed your two major concerns by adding an experimental evaluation and clarifying the limitations (and purpose) of working in the purely spurious setting.  \n\n> There are no experiments with either simulated or real-world data. At a minimum, it would be beneficial to include simulated experiments that verify the basic claims regarding the optimality of the counterfactually-fair predictors and the relationships between counterfactual fairness and the observational fairness criteria under different data generating processes.\n\nAs suggested, we have added an experiment to verify the main claims of the paper. See the global response for details. We find that on a (semi-synthetic modification of) the Adult income dataset: \n1. The counterfactually fair predictor trained on biased domain data has good performance when deployed on the unbiased target distribution (Theorem 1).\n2. The counterfactually fair predictor satisfies the observational fairness criteria predicted by the underlying causal structure (Corollary 2.1).\n\n> The restriction of the scope of the work to the purely spurious setting (line 201) limits the generalizability of the claims and this is largely not acknowledged. The work makes the claim that the purely spurious setting where association between group attributes and outcomes are due to selection is reasonable to cover the space of problems where fairness is of interest. I would argue that this is a serious limitation of the work because of the aim to provide guidance that broadly applies to the space of settings where it is of interest to evaluate fairness and bias considerations.\n\n> For an example where the spurious setting may not hold ... this setting does not map cleanly on to the type of selection graphs considered in this work.\n\n> It would be beneficial for this work to expand upon the potential limits of the generalizability of its claims, especially due to the assumption of the purely spurious setting. Relatedly, it is not clear how well the claims are applicable outside of the set of graphs and selection mechanisms considered in this work.\n\nWe see that this is an important limitation and appreciate the reviewer’s critique. We discuss this in the global response, but to make the purely spurious assumption more prominent in the paper, we have added it to the Introduction and to the Discussion (the final section). We have also elaborated on it on Page 6 to more fairly and thoroughly present the pros and cons when we formally introduce it. \n\nAs discussed in the global response, assuming pure spuriousness substantially simplifies the results (and thus also the presentation), allowing us to thoroughly develop the connection from counterfactual fairness to out-of-domain robustness and observational fairness metrics. However, the purely spurious assumption is likely not necessary to make such a connection in general. We note that extending the results to more complicated causal scenarios and fairness definitions is an important direction for future work. (However, we think that the extra level of technical complexity would severely obfuscate the core point in the present paper).\n\n> There is inconsistency in which variable is used to the ground truth outcome versus the noisy outcome with measurement error. In lines 112-113, Y-tilde is considered the noisy label and Y the ground truth. Elsewhere in the paper, Y-tilde is considered the ground truth.\n\nWe have made $\\tilde{Y}$ the ground truth throughout.\n\n> To what extent should the statements about the observable implications of counterfactual fairness be considered if and only if statements versus unidirectional statements? For instance, under Corollary 2.1.1 does demographic parity imply counterfactual fairness under the causal measurement error graph?\n\nObservational data can strongly constrain causal structure but usually cannot uniquely identify it. Accordingly, we do not expect the results to be bidirectional in general. However, with the purely spurious assumption and faithfulness, the relationships seem bidirectional.\n\n> Could you please add additional detail to prove the result regarding predictive parity? ... It is also unclear whether the argument assumes that score calibration implies predictive parity…\n\nThis is an error. Statement 2.2.6 should say “with positive outcome label (i.e., $D=1$)” instead of “with positive outcome (i.e., $Y=1$)”. We sincerely apologize for the mistake and the confusion. Please let us know if you would still like additional detail, but hopefully that clarifies the statement and proof. \n\nRegarding score calibration, if a score-calibrated predictor were converted into a binary predictor by using a score threshold, it would not necessarily imply predictive parity—as you mention is explained in Chouldechova (2017). The corrected version of Statement 2.2.6 does not mention score calibration, nor does its proof (lines 601-602), so we currently don’t believe the argument assumes this. Note that the proof only refers to binary calibration, which does not have the issue that Chouldechova raises in which $S | R = r$ (i.e., $S | A = a$) potentially differs across groups in ways that result in PPV imbalance.\n\n> Minor suggestions:\n> * Introduce the notion that a counterfactually fair predictor depends only on X_{A}^{\\perp} in the main text rather than in the > supplementary material\n> * Replace “correlational” with “observational”\n> * Replace “persecuted” with “prosecuted”\n\nDone."}]

Author rebuttal:
We thank the reviewers for their time and thorough commentary. We are glad to hear the core technical results were seen as valuable contributions (“interesting way to connect…”, “technically sound”, “useful”, “novel”, “the theoretical results are of high quality”, “expand upon the prior work meaningfully”) and the general presentation was clear (“integrating several literatures into a coherent frame”, “clear”, “really good job making the point”, “generally well-written and clear”).

There were two main reviewer concerns: experimental evaluation of the theoretical results and real-world applicability.

# Experiments
As several reviewers suggested, we conducted an experimental test of the main theoretical predictions of the paper. See the attached PDF for the table of results. Per NeurIPS policy, we have sent a code link to the AC privately.

As predicted, we find:
1. A counterfactually fair predictor satisfies the observational fairness metric corresponding to the underlying causal graph (Corollary 2.1).
2. Counterfactually fair predictors trained on biased data (where the protected class is associated with the outcome) have strong predictive performance on the unbiased target domain where no association exists (Theorem 1).

To conduct this experiment,
1. We produce a novel counterfactually fair (CF) predictor for the purely spurious setting, $f_{CF}(X) = P(Y=1|X, A=1)P(A=1) + P(Y=1|X, A=0)P(A=0)$ (i.e., a weighted average of naive predictors across each level of the protected class). This follows because $P(Y=1| X, A) = P(Y=1|X_A^\perp,A)$ in the spurious setting. Then $f_{CF}$ only depends on $X_A^\perp$, so it is in fact CF. $f_{CF}$ can be readily estimated in practice. Namely, fit a model $f_{naive}(X,A)$ that predicts $Y$ from both $X$ and $A$ (thus estimating $P(Y=1 | X, A)$) and then let $f_{CF}(X) = f_{naive}(X, 1) P(A=1) + f_{naive}(X, 0) P(A=0)$.
2. We use the Adult income dataset to construct semi-synthetic data matching each of the causal graphs. We first construct a synthetic binary protected class $A$, sampled at random. We then have $A$ causally affect $X$ by fixing the value of several attributes with a fixed probability when $A=1$ (if $A=0$, no change is made). We then create three biased datasets corresponding to the three causal graphs shown in Figure 2 by (a) noising $Y$ according to $A$, (b) selecting data points based on $A$ and $Y$, and (c) selecting on $A$ and predictors of $Y$.
3. We train CF predictors on each biased dataset then check the performance of the CF predictor on the base data where $A$ and $Y$ are independent (Theorem 1) and check observational fairness metrics (Corollary 2.1).

# Real-world connections and the ‘purely spurious’ assumption
The other concern that reviewers raised is about real-world applicability. In particular, there is a concern that the ‘purely spurious’ requirement is too strong in practice and is not sufficiently acknowledged as a key limitation.

We basically agree with this point! We view the contribution of the paper as novel technical results that show counterfactual fairness is both measurable and desirable in some natural situations. The significance of this, which the reviewers appreciated, is:
1. Counterfactual fairness is an important notion socially (e.g., it maps to a legal notion of discrimination).
2. It’s not obvious a priori that counterfactual fairness can ever be assessed or connected to other observational fairness metrics. We show that considering causal structure can allow this.
3. Prominent existing work [1] suggests that counterfactual fairness results in an unacceptable penalty to performance. Our results on out-of-domain performance directly rebut this by showing counterfactual fairness can be desirable even if one is concerned only with performance.
Focusing on the purely spurious setting lets us make these points clearly, avoiding obfuscation from extra technical complexity. We follow the recent literature in using this assumption [e.g., 2,3,4] to make meaningful progress in this challenging area.

We also think it should be possible for future work to move beyond the purely spurious case. There is a literature on causal fairness beyond counterfactual fairness that offers guidance on how to define fairness notions in more complex scenarios [e.g., 5, 6]. Given an assumed causal structure and a desired causal fairness notion respecting this structure, one should be able to extend these results to articulate the kind of domain shifts where a fair predictor is expected to be robust and to derive observable signatures of the fairness notion. We do not believe we have room in a single paper to thoroughly develop the core technical results and also tackle these complexities, but this is an exciting direction for future work.

As suggested by the reviewers, we have also changed the text to foreground the purely spurious assumption as a limitation. In response to comments that the paper would be stronger with real-world application (e.g., “if the pipeline of Figure 1 were carried out to show the steps tangibly”), we have conducted the experimental test but also foregrounded the role that we see these results playing in that broader pipeline.

[1] Nilforoshan, Gaebler, Shroff, and Goel. 2022. “Causal Conceptions of Fairness and Their Consequences.” PMLR. doi:10.48550/2207.05302.

[2] Makar and D’Amour. 2023. “Fairness and Robustness in Anti-Causal Prediction.” TMLR. doi:10.48550/2209.09423.

[3] Makar, Packer, Moldovan, Blalock, Halpern, and D’Amour. 2022. “Causally Motivated Shortcut Removal Using Auxiliary Labels.” AIStats. doi:10.48550/2105.06422.

[4] Veitch, D’Amour, Yadlowsky, and Eisenstein. 2021. “Counterfactual Invariance to Spurious Correlations in Text Classification.” NeurIPS. doi:10.48550/2106.00545.

[5] Plecko and Bareinboim. 2022. “Causal Fairness Analysis.” doi:10.48550/2207.11385.

[6] Nabi, Razieh, and Ilya Shpitser. 2018. “Fair Inference On Outcomes.” AAAI. doi:10.1609/aaai.v32i1.11553.

Meta-review: The paper connects counterfactual fairness with correlational fairness and explores its relationship with out-of-distribution generalization. The approach is interesting, technically sound, and the paper is well-written. There were some questions and concerns raised by reviewers including the connection to impossibility theorems, the assumption of decomposing variables, and the connection between title and paper's actual content. The paper would benefit from addressing these comments.
Decision: Accept (poster)