Causal Fairness under Unobserved Confounding: A Neural Sensitivity Framework
Authors: Maresa Schröder
Dennis Frauen
Stefan Feuerriegel
Venue: ICLR 2024 poster (2024)
Keywords: Causal Fairness
Sensitivity Analysis
Fair Prediction
Unobserved Confounding
Primary area: societal considerations including fairness, safety, privacy

Abstract: Fairness for machine learning predictions is widely required in practice for legal, ethical, and societal reasons. Existing work typically focuses on settings without unobserved confounding, even though unobserved confounding can lead to severe violations of causal fairness and, thus, unfair predictions. In this work, we analyze the sensitivity of causal fairness to unobserved confounding. Our contributions are three-fold. First, we derive bounds for causal fairness metrics under different sources of observed confounding. This enables practitioners to audit the sensitivity of their machine learning models to unobserved confounding in fairness-critical applications. Second, we propose a novel neural framework for learning fair predictions, which allows us to offer worst-case guarantees of the extent to which causal fairness can be violated due to unobserved confounding. Third, we demonstrate the effectiveness of our framework in a series of experiments, including a real-world case study about predicting prison sentences. To the best of our knowledge, ours is the first work to study causal fairness under observed confounding. To this end, our work is of direct practical value for auditing and ensuring the fairness of predictions in high-stakes applications.

Published as a conference paper at ICLR 2024
CAUSAL FAIRNESS UNDER UNOBSERVED CONFOUNDING: A NEURAL SENSITIVITY FRAMEWORK
Maresa Schröder, Dennis Frauen & Stefan Feuerriegel
Munich Center for Machine Learning
LMU Munich
ma masa. sc schroeder, frauen, frauenn,euurrriieeeeeee .de
ABSTRACT
Fairness of machine learning predictions is widely required in practice for legal, ethical, and societal reasons. Existing work typically focuses on settings with- out unobserved confounding, even though unobserved confounding severe violations of causal fairnsss and, thus, unfair predictions. In this work, we analyze the sensitivity of causal faimess unobserved confounding. Our contri- butions are three-fold. First, we derive bounds for causal fairness metrics under different sources of unobserved confounding. This enables practitioners to exam- ine the sensitivity of their machine learning models to unobserved confounding in fairness-critical applications. Second, propose novel neural framework for learning fair predictions, which allows us to offer worst-case guarantees of the extent to which causal fairness can be violated due unobserved confounding. Third, we demonstrate the effectiveness of our framework in a series of experi- ments, including real-world case study about predicting prison sentences. To the best of our knowledge, ours is the first work to study causal fairness under unobserved confounding. this end, our work is of direct practical value as a refutation strategy to ensure the fairness of predictions in high-stakes applications.
lead to
INTRODUCTION
Fairness of machine learning predictions is crucial to prevent harm to individuals and society. For this reason, fairness of machine learning predictions widely required by legal frameworks (Baro-
cas & Selbst, 2016 De-Arteaga al. 2022 Dolata al. 2022; Feuerriegel et al.) 2020 Frauen
et al.] 2023a Kleinberg et al. 2019). Notable examples where fairness is commonly mandated by law are predictions in credit lending and law enforcement.
A prominent fairness notion is causal fairness (e.g..
Offerie, drug type
Kilbertus al.) 2017 Plecko & Bareinboim 2022
Zhang & Bareinboim 2018b). Causal fairness lever- ages causal theory to ensure that given sensitive attribute does not affect prediction outcomes. For example, causal fairness, prison sentence may vary by the type of offense but should not be affected by the defendant's race. Causal fairness has several advantages in practice as it directly relates to legal terminology that outcomes must be independent of sensitive attributes (Barocas & Selbst 2016)
Race
Crime history
Family backg
Seratiw atriott
Bitholace
Octsrss Unbboweed cortorrder.
Figure 1: Example: Causal graph for predict- ing prison sentences.
Existing works on causally fair predictions (c.g.. Nabi & Shpitser 2018 Zhang & Bareinboim, 2018a b) commonly focus settings with un- observed confounding and thus assume that the causal structure of the problem fully known. However, the assumption of unobserved confounding is fairly strong and unrealistic in prac-
In the literature, there is some ambiguity in the terminology around causal faireess. By referring to our fair- ness notion as "causal fairness". we are consistent with previously established literature (Plecko & Bareinboim 2022). However, acknowledge that there other fairesss notions hased causal theory (e.g., counterfac- faimess), which do present the focus here. More details about the difference in Supplement

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approach and model different sources of unobserved confounding (e.g., between Published as a conference paper at ICLR 2024
tice (Carey & Wu 2022 Fawkes al. 2022 Kilbertus et al. 2019 Loftus et al. 2018: Plecko & Bareinboim 2022). This can have severe negative implications: due to unobserved confounding, the notion of causal fairness may be violated, and, result, predictions can cause harm as they may be actually unfair.
An example is shown in Fig. Here, a prison sentence (=outcome) should be predicted while counting for the offense and crime history (=observed confounders). Importantly, race (=sensitive attribute) must not affect the sentence. However, many sources of heterogeneity are often missing in the data, such as the housing situation, family background, and birthplace (=unobserved con- founders). If not accounted for, the prediction of prison sentences can be biased by the confounders and, therefore, unfair. Additional examples from credit lending and childcare are in Supplement B.1
Our paper: We analyze the sensitivity of causal fairness to unobserved confounding. Our main contributions are three-fold (see Table 1
1. We derive bounds for causal fairness metrics under different sources of unobserved con- founding. Our bounds serve as refutation strategy to help practitioners examine the sen- sitivity of machine learning models to unobserved confounding in fairnes--crttical applica- tions.
2. We develop a novel neural prediction framework, which offers worst-case guarantees for the extent which causal fairness may be violated due to unobserved confounding.
3. We demonstrate the effectiveness of our framework in experiments synthetic and real- world datasets. To the best of our knowledge, we are the first to address causal fairness under unobserved confounding.
Literature stream
Fair causal effects (2019a)
Unobs, confounding
Level
2 RELATED WORK
Causal faimnes
Zrang Rareinboim (2018a)
x Freuen al (2020b). (2006)
Fairness notions based on causal rea- soning: Fairness notions based on causal reasoning have received increasing atten-
Sensitivity analysis
Our work
3
tion (c.g., Chiappal 2019
Chikahara et al.
2021
Huang al. 2022
Khademi et al.
Table 1:
We make contributions to multiple literature streams. Level according Pearl's 's causality ladder.
2019
Kilbertus et al.] 2017 2011
2019: Nabi &
Shpitser 2018
8 Nabi al.]
2019 Wu et al.
2019a b 2022 Yao et al. 2023 Zhang et al.
2017 Zhang & Bareinboim 2018a b). For a detailed overview, we refer (Loftus et al. 2018 Nilforoshan et al.) 2022 Plecko & Bareinboim 2022).
Different fairness notions have emerged under the umbrella of causal reasoning (see Supplement C for details). For example, the notion of counterfactual fairness (Kusner et al. (017)) makes strictions on the factual and counterfactual outcomes. A different notion is causal fairness (Nabi & Shpitser 20188 Plecko & Bareinboim] 2022). Causal fairness makes restrictions on the causal graph. It blocks specific causal pathways that influence the outcome variable and are considered unfair. In this paper, we focus on the notion of causal fairness.
Fairness bounds in non-identifiable settings: One literature stream (Nabi & Shpitser 2018 Wu jet al. al. 2019b) establishes fairness bounds for settings that are non-identitiable due to various reasons. While we also derive fairness bounds, both types of bounds serve different purposes (i.e., to address general non-identifiability vs. specificity to unobserved confounding).
Fair predictions that are robust to miss-ppecification: The works by Chikahara et al. (2021) and Wu et al. (2019a) aim to learn prediction models that ensure fairness even in the presence of model misspecification. However, none of these works trains prediction models for causal faimess that are robust to unobserved confounding. This is one our novelties.
Sensitivity of fairness to unobserved confounding: To the best of our knowledge, only work analyzes the sensitivity of fairness to unobserved confounding (Kilbertus et al. 2019), yet it has crucial differences: (1) Their paper focuses on counterfactual fairness, whereas focus causal fairness. (2) Their paper allows for unobserved confounding only between covariates, while we fol- low
Both data and code for framework available in our GitHub repository.
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covariates and sensitive attribute). (3) Their paper builds upon non-linear additive noise models. We do not make such restrictive parametric assumptions.
Sensitivity models:
Sensitivity models J.g..nesson et al. 2022 Rosenbaum 1987 Tan 2006
common tools for partial identification under unobserved confounding. We later adopt sensitivity setting. However, it is not possible to directly apply sensitivity models
models to formalize to our setting (see Table The reason is that sensitivity models perform partial identification on interventional effects (-level 2 in Pearl's causality ladder), whereas causal fairness involves (nested) counterfactual effects (=level 3). For an in-depth discussion, see Supplement C.3
Research gap: To the best of our knowledge, no work has analyzed the sensitivity of causal fairness to unobserved confounding. We are thus the first to mathematically derive tailored fairness bounds. Using our bounds, we propose novel framework for learning fair predictions that are robust the misspecification of the underlying causal graph in terms of unobserved confounding.
3 SETUP
Notation: We use capital letters to indicate random variables X with realizations I. We denote probability distributions over by Px, i.e., X ~ Px. For ease of notation, we omit the subscript whenever it is clear from the context. If X is discrete, we denote its probability mass function by P(x) P(X- = x) and the conditional probability mass functions by P(y x)) = P(Y = X a) for a discrete random variable Y. If X is continuous, p(æ:) then denotes the probability density function w.r.t. the Lebesgue measure. counterfactual outcome to y under intervention E denoted ya- We denote vectors in bold letters (e.g., X). For summary of notation, see Supplement A
We formulate the problem based on the structural causal model (SCM) framework (Pearl 2014).
Definition 1 (SCM). Let V (Vi...,, .i..... Vn denote a set of observed endogenous variables, let U Pu denote a set of unobserved exogenous variables, and let F = L with fv, : Pa(V)) VUU- > V, The tuple (V, U,F, Pu) is called a structural causal model (SCM).
We assume directed graph Gc induced by SCM C to be acyclic.
Setting: We consider the Standard Fairness Model (Plecko & Bareinboim 2022). The data V =A, Z, Z, consist of a sensitive attribute E A, observed confounders Z Z, mediators M E M, and outcome Y E IR.
Z
Senstthve attribuue Dutcome varablles Unobuereed
U
U
Unique to our setup that we allow for three types of unob- served confounding in the model: (1) unobserved confounding between A and Y. which we refer to as direct unobserved con- founding UDE; (2) between A and M, called indirect unob- served confounding UIg; and (3) between A and Z, which we refer to as spurious confounding USE. The set of unobserved confounders is then given by U = (UDE, UIE USE Fig. 2 shows the causal graph of the Standard Fairness Model, where we show the different sources of unobserved confounding.
Y
U
Figure 2: Causal graph of the Stan- dard Fairness Model with different sources of unobserved confound- ing UDE, UIE and USE.
For ease of notation, we focus on discrete variables Z, M and binary A, where for simplicity |A| M
= 1. This is common in the literature (c.g.. Khademi Wu al. 2019a). Further, it also matches common applications of our framework in law enforcement (Fig. []. as well as credit lending and childcare (Supplement B.I). Supplement [
Z
al. 2019 Plecko & Bareinboim, 2022;
we discuss the extension to continuous features and provide
additional experimental results.
Path-specific causal fairness: Following Zhang Bareinboim 2018a). we define the following path-specific causal fairness as: (1) counterfactual direct effect (Ctf-DE) for the path A + Y; (2) indirect effect (Ctf-IE) for the path A + M + Y; and (3) spurious effects (Ctf-SE) for the path Z A.
Ctf-DE measures direct discrimination based on the sensitive attribute. Ctf-IE and Ctf-SE together measure indirect discrimination through factors related to the sensitive attribute, potentially working as a proxy for the latter. The Ctf-IE corresponds to all indirect causal paths, whereas the Ctf-SE measures all non-causal paths. In the prison sentence example (Fig. I.. the Ctf-DE could measure
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unequal prison sentences based on race. The Ctf-IE could be driven by the defendant's court verdict history, which itself might have been illegitimately affected by the offender's race.
Discrimination in terms of causal fairmess is formulated through nested coun- terfactuals for each of the above path-specific causal fairness effects. Formally, one defines the discrimination of realization A a, compared A aj for each path conditioned realization
Causal fairness:
A = a, i.e,
DE (y a) == P(ya, ma:
a) - P(ya, la),
IBa,a, (y) a) = Pay a) P(ya, I a),
(2)
SE (V) = P(yo, | [0,) Ply a.). The total variation, i.c., the overall discrimination of individuals with A = compared to A = can then be explained as TVa.a, DE,,,,, BBareinboim 20)8 Causal fairness is achieved when Da (y a), IFag, (y a). Sta,,ay (8) are zero or close to zero. In practice, this typically operationalized by enforcing that the effects are lower than user-defined threshold. For ease of notation, we abbreviate the three effects as DE, IE, and SE. We further use CF to refer to any of the above three effects, i.e., CF (DE, IE, SE}. We note that, although we focus on the three effects specified above, framework for deriving fairness bounds can also be employed for other fairness notions (see Supplement (F).
(3)
aa) 1E, (0 (y) a:) SEa,, (y) (Zaag &
Our aim: We aim to ensure causal fairness of prediction model in settings where there exists unobserved confounding, i.e., UDE UIE USE F O. Existing methods (e.g.. Khademi et al. 2019. Zhang & Bareinboim| 2018a) commonly assume that unobserved confounding does not exist, i.e., UD, UIE USE = O. Therefore, existing methods are not applicable to our setting. Hence, we first derive novel bounds for our setting and then propose a tailored framework.
4 BOUNDS FOR CAUSAL FAIRNESS UNDER UNOBSERVED CONFOUNDING
In the presence of unobserved confounding, the different path-specific causal fairness effects - i.e., CF E (E, SE} are not identifiable from observational data, and estimates thereof are thus biased (Pearl 2014). As a remedy, we derive bounds for DE, IE, and SE under unobserved con- founding. To this end, we move from point identification to partial identification, where estimate bounds under a sensitivity model S (elaborated later). Formally, we are interested in an upper bound C and lower bound CF5s corresponding to the path-specific causal fairness effect CF for all CF E (DF, SF..
To do so, we proceed along the following steps (see Fig. 3 (DE, IE, SE} into identifiable parts and unnested interventional effects. 2 We then use suitable sensitivity model S and the interventional effects to derive the upper CF and lower bound CF- for each CF E (DE, IE, SE}. Building upon our bounds, we later develop a neural framework for learning causally fair prediction model that is robust to unobserved confounding (see Sec. 5)
1 We decompose each CF E
Serail wity model
Countiofi unisiing
Furtil dlcntiottionn
Constraine optinizatinn
Unneated CP
Palmess bounds
Feir prepirrcctiono
potoncally nosted
donnd Sabie and untoiillitl terrs
deviiiibbl to noun seralliity analyats
contrie maxdmmm farrisss
Figure 3: Our workflow for deriving bounds on causal fairness under unobserved confounding. 4.1 SETTING: GENERALIZED MARGINAL SENSITIVITY MODEL
We derive fairness bounds in the setting of causal sensitivity analysis (Imbens) 2003 Rosenbaum 1987).
Here, the underlying idea is relax the assumption of unobserved confounders by allow-
Nested counterfactuals express multi-layered hypothetical scenarios by considering counterfactual state- ments of subsequent events such as A + M Y. One can view nested counterfactuals multi-intervention counterfactuals in which one intervention is (partly) intervened on by second one, e.g., P(ya,, (,,)). ). which reads as the probability of outcome Y y under the intervention A and the intervention M m where itself is not affected by intervention A a, but A az.
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ing for a certain strength of unobserved confounding through so-called sensitivity models. In our work, we adopt the generalized marginal sensitivity model (GMSM) (Frauen et al. 2023b). A key benefit of the GMSM is that it can deal with discrete mediators and both discrete and continuous out- comes. Importantly, the GMSM includes many existing sensitivity models and thus allows to derive optimality results for broad class of models. For an overview of causal sensitivity models and discussion of their applicability to derive fairness bounds, we refer the reader to Supplement C.2
Definition 2 (Generalized marginal sensitivity model (GMSM)). Let V = A. Y) Let A, M and Z denote set of observed endogenous variables, U set unobserved exogenous variables, and G a causal directed acyelic graph (DAG) on VJU. For an observational distribution Py on V and a family P of joint probability distributions on V UU that satisfy
P(U = uw 2,0)
(4)
(1-Tw) 2) +Iw
P(U uw 2, do(())
Tw p(a 2) +
for W E (M,Y), the tuple = (V, Pv, is called a weighted generalized marginal sensitivity model (GMSM) with sensitivity parameter I'w 2 1.
The sensitivity parameter I' I'w controls the width of the interval defined by the bounds. In practice, it is determined based on domain knowledge of the magnitude of unobserved confounding on the association between W and A or through data-driven heuristics (e.g.. Jin et al., [223] Kallus et al.) 2019). Importantly, our framework applies not only for known ww but also of practical value for unknown I W. We discuss practical considerations for both cases in Supplement B.2
We introduce further notation: An acyclic SCM = Uc, F, Pu), with S Uc. 9 S 9c, compatible with sensitivity model S if Uc does not introduce additional unobserved confounding and the probability distribution Pvote induced by C belongs to P, i.e., Pvuue E The class of SCMs C compatible with S is denoted as K((S).
Objective for bounding path-specific causal fairness: We now formalize our objective for iden- tifying CF3 and CF For CFc E {DE, IE, SE), we aim to find the causal effects that maximize (minimize) CF& over all possible SCMs compatible with our generalized sensitivity model S, i.e.,
CF
sup CFc CEK(S)
and
CF
inf CFc. CEK(S)
(5)
As a result, we yield upper (+) and lower (-) bounds DE JE and SE for the different effects.
4.2 COUNTERFACTUAL UNNESTING (STEP) 1)
The gath-speciffe causal fairness effects CFc E {DE, SE) contain nested counterfactuals (=level 3 of Pearl's 'ss ladder of causality). Sensitivity analysis, however, is built upon interventional effects (=level 2 of the ladder). Thus, incorporating sensitivity analysis into the fair prediction framework is non-trivial. First, we must decompose the expressions CF into identifiable parts and interventional effects, which solely depend on one intervention at a time. Hence, we now derive an unnested expression of CFcc in the following.
Lemma 1. CFc (DE, IE, SE} can be defined as monotonically increasing function h over sum of unidentifiable effects e E E in the SCM C, where denotes the set of all effects in C, and an identifiable term i.e.,
CF - h ( E((v,C)+g) for CFc E (DE, IE, SEJ.
330
Proof. The result follows from the ancestral set factorization theorem and the counterfactual factor- ization theorem in (Correa et al. 2021). A detailed proof is in Supplement D
'Other sensitivity models from the literature such as the marginal sensitivity model (MSM) Tan 2006
special cases of the aforementioned and therefore only of limited use for general setting (c.g.. the MSM can only handle sensitive attributes that are binary but discrete, they can estimate path-specific causal fairness). practice, it is up to the user to replace the GMSM with suitable sensitivity model our framework.
For a rigorous definition, see Supplement
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The effects represent the single-inter interviintevinnnnnn counterfactual effects, which can be bounded through causal sensitivity analysis. The term & can be directly estimated from the data. Now, using Lemma can rewrite Eq. (5) in unnested way.
Remark 1. Let Ex be the of inteeeiintevvnnionn counterfactual effects with e(v, S) 2 0 and E the set of e with e(v,S) < 0. Then, the bounds for CF E (DE, IE, SE} under GMSM S in Eq. (5) can be obtained as
CF'S = 1 E
sup c(v,C)- CEX(S)
E
inf CER(S)
ce(v,) l ()
(7a)
CESt
CEE
CF3 = (
Σ
inf e(v,C) CEX(S)
Σ
sup CEK(S)
eevv,) +9).
(7b)
eE81
EEE
4.3 SENSITIVITY ANALYSIS TO BOUND PATH-SPECIFIC CAUSAL FAIRNESS (STEP 2)
We now use the expressions from counterfactual unnesting to derive upper (+) and lower (-) bounds for the three path-specific causal effects (i.e., DE IE+, and SE*) under unobserved confounding (U- {UDE, ULB DSE} * O). The following theorem states our main theoretical contribution.
Theorem (Bounds on path-specift causal fairness). Let the sensitivity parameters for M and Y in GMSM be denoted by I' a' and ly. For binary sensitive attribute A E {0, 1}, where, for simplicity, we focus on discrete Z, the individual upper (+) and lower (-) bounds on the path-specific causal fairness effects for specific y. azs aj are given by
DE ( [a)a P(a:)
Σ p* (y m,,lm,,,,, (
ZEZ,
mEM
(8)
P(aj)
Σ Ply
Ply a.),
P(a.)
ZEZ. mEM
P(a.)
IE B (vlo,)
l Piylm,,,a))Ppr 2, P(m P.o;))((ss)
P(a;)
ZEZ. mEM
P(z)
p ( 22,,)
(9)
P(ay)
#E 7
mEM
(y m, (n, (m z,
P(ai) P(a;)
SEa, (v)
(m 2. P(s) (m) 1
Ply I [a))
(10)
P(a3)
=EZ.
mEM
for ai, aj E A. Note that, in the continuous case, the sum over Z has to be replaced by the integral. For discrete M with (mm) denoting the CDF of P(m z, az), we define
P(m! - 1 +))+
if F(m) < 114 TM
-Tr
P(m
+Ta)
F F(m -1) 1-1'24"
p+ pmmm| 12,a,)
((1- Pxoo 1-m-1((m))))
14TH
+P(a (P(m) F(m 1)(1 FT))) )) otherwise,
P(m | z, (1- 2) +2)
if Ff < 1+fM if F((m 1) > I+TM
P(a: P(m | 2, (1) - | P(a: 22) +
p- 22);))
(--Ma))))(-aaa) + P(ayls) + F[mm] F(m - 1)TM
(14F
+P(as =)(F(m)) F(m I)))
otherwise.
Y is discrete, the probability functions pt Pt m, 2, 04) and P (y m, 2, a;) are defined analo- gously. The probability functions for continuous Y are presented in Supplement F

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Proof. We prove the theorem in Supplement D The proof proceeds as follows: We unnest each counterfactual term in DE, IE, and SE to receive expressions of the form given in Eq. (6) Subse- quently, derive bounds for each unidentifiable part in the unnested counterfactuals through sen- sitivity analysis. Finally, we combine the bounds to prove the above statement for DE, IE, SE. I7
The bounds provide worst-case estimate for CF E {DE, IE, SE} for specific y, ai aj under unob- served confounding. Training predictors based on the bounds reduces the risk of harmful and unfair predictions due to violating the no unobserved confounding assumption. The worst-case fairness estimates are guaranteed to contain the true path-specific causal effects for sufficiently large sen- sitivity parameters. The wider the interval defined through CF+ and CF-, the more sensitive the path-specific causal effect to unobserved confounding, i.e., the less confidence we can have in our results not accounting for unobserved confounding. Thus, one can test the robustness of prediction models as to what extent they are sensitive to unobserved confounding in high-risk applications.
We emphasize that Theorem is independent of the distribution and dimensionality of unobserved confounders and specific SCM formulations. Hence, the bounds are valid for discrete, categorical, and continuous outcome variables, making them widely applicable to real-world problems.
Remark 2. Our bounds are sharp; that is, the bounds from Theorem D Mrre optimal, and, therefore, the equality sign in Eq. 5 holds.
The remark follows from (Frauen et al. 2023b). Our bounds can be interpreted as the supremum or infimum of the path-specifi causal effects in any SCM compatible with the sensitivity model.
BUILDING A FAIR PREDICTION MODEL UNDER UNOBSERVED CONFOUNDING
Our goal is to train a prediction model So that is fair under unobserved confounding, where fairness denoted by the path-specific causal fairness effects. Recall that path-specific causal fairness is unidentifiable under unobserved confounding. Therefore, we use our bounds CF+ and CF for CF E (DE, IE, SE} from Theorem I to limit the worst-case fairness of the prediction, where worst- case refers to the maximum unfairness, i.e., maximum deviation from zero, of each of the path- specific causal effects under a given sensitivity model. Intuitively, we want our prediction model to provide both accurate predictions and fulfill user-defined fairesss constraints for each path-specific causal effect, allowing the practitioner to incorporate domain knowledge and business constraints.
Bound computation: Theorem 1 requires bounds p* (y 2, m,) and p*(m z,a) to calcu- late the overall bounds CF+, CF for CF E {DE, IE, SE} for specific y, a2 az. For training a robustly fair prediction model, we are now interested in deriving effects CF (a;, az) of the form DE a (E[Y] | a;) and similarly for IE and SE As a result, we replace p- (y 2, m, a) by the non-random predicted outcome fa(a, 2, m). This way, we can estimate the unidentifiable effect p* (y 2, m. a) and only need to obtain p* pmmm 2, a). This requires the conditional density esti- mation for P(a 2). P(m a) through pre pre-trained estimators 9A Z + A and gar A, Z + M.
Objective: Our objective for training the prediction model fe consists of two aims: (1) low prediction loss I(fo), and (2) worst-case faimess estimates bounded by user-defined threshold
7 (DDE; ME, 7SE]" We formulate our objective as a constraint optimization problem:
min I(fo) s.t. max (as, ICF (a;, a; & TCF, CF E (DE, IE, SE), 0
(11)
To solve Eq. (11) by gradient-based solvers, we first need to rewrite the constrained optimization problem to single loss function. For this, we make use of augmented Lagrangian optimization Nocedal & Wright, 2006). which allows us to add an estimate À of the exact Lagrange multiplier to the o We provide pseudocode for our training algorithm in Algorithm I In each iteration, we calculate the upper ub(-) and lower bound lb(-) on each CFg(a, az). CF E [DE, IE, SE} according to Theorem D In Supplement B we provide additional explanations and an extended algorithm for multi-class classification.
'For regression and binary classification, the expectation is reasonable choice of distribution functional. For multiclass classification, replacing the expectation with other distributional quantiles or constraining the effects for each class separately is also common in practice. More details are in Supplement E.4 For detailed description; Chapter 17 in Nocedal & Wright (206)..
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Implementation details: We imple- ment the underlying prediction model three-layer feed-forward neu- ral network with leaky ReLU acti- vation function and dropout. Our framework additionally requires pre- trained density estimators gA 9M, for which we also use a feed-forward neural network. Details about the model architectures and hyperparam- eter tuning are Supplement [ in which we also report the performance of our prediction models.
Algorithm 1: Training fair prediction models robust
unobserved confounding
Input: Duia ((A,, Z., Mr. i (1. ) sensiavity parameter M' vectur of kammes constraints Lagrangium purameter vecturs Ao and estimatorss
Po.
inital prediction model fog.
convergence criterion e, Lagranzian update rate
Ouigut: Fuir predictions (Di 1 E (l. n iniine prediction model fg -Mk Pgg
Ie;
J8g A
nentions
/4 Trair prediction model
neseed spects da
5 + Saj (A, Z, M); Detwenine CPR. CFT for CP E (DE, IE,
rclieving Throron B
CF + ublfe (A, F.). 8A 9Ar);
CFR + Ib[fe, (A, 2, FM, 9M):
is optimization objective tollowing Eg. for CP (DR, IR, SR) de
ccp + max (CF (I.
6 EXPERIMENTS
Baselines: We emphasize that base- lines for causal fairness under unob- served confounding are absent. We thus compare the following models: (1) Standard refers to fo trained only on loss I without causal faireess straints. Hence, the standard model should lead to unfairness if trained on an unfair dataset, even without unob- served confounding. (2) Fair naïve is prediction model where the loss additionally aims to minimize unfair- ness in terms of the unconfounded path-specific effects. Hence, causal fairness is considered only under the assumption of no unobserved confounding. (3) Fair robust (ours) our framework in Al- gorithm in which we train the prediction model wrt. causal fairness while additionally accounting for unobserved confounding. Importantly, the architectures of the neural network fa for the different models identical, fairness improves only due
(eDR: "sRl"
lagrangian lossl (o,) ) The (Ax Ac-1)
Update paranstera predictos
10
_ + optimlserr(agrragiaan fo,)
11
end
12
if
7 then Check convesyence if prediction (Ss)) S. then
13
14
10
16
and
17
end
Update Legrangian paraneters
*/
18
A (A, CHt, O. -M---1 apt
19
end
tailored learning objective.
We report path-specific causal fairness by computing bounds on the three effects following Theorem [ Idcally, the different values remain zero, even in the presence of unobserved confounding. We report the averaged results and the standard deviation over five seeds. We later report results for different sensitivity parameters F in the Supplements.
Performance metrics:
6.1 DATASETS
Synthetic datasets: We follow common practice (e.g.. |Fraeen eet al. 2023b; Kusner et al.] 2017) building upon synthetic datasets for evaluation. This has two key benefits: (1) ground- truth outcomes are available for bench-marking, and (2) we can control the level of unobserved confounding to understand the robustness of our framework.
UDB
Cantur
Diear ptrie
UIB
Figure 4: Validity of our bounds. Our bounds successfully con- tain the oracle effects for different confounding levels T. Results for different sensitivity parameters M and Ty 1.
We consider two settings with different types of unobserved confounding. Both contain single binary sensitive attribute, me- diator, confounder, and outcome variable. In setting "UDE". we introduce unobserved confounding on the direct effect with confounding level & and in setting "UlE". unobserved confounding on the indirect effect with level &b We generate multiple datasets per setting with different confounding levels, which we split into train/val/test sets (60/20/20%) Details are in Supplement(H..
Real-world dataset: Our real-world study is based on the US Survey of Prison Inmates (United
States. Bureau of Justice Statistics 2021). We aim to predict the prison sentence length for drug
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offenders. For this, build upon the causal graph from Fig. We consider race as the sensitive attribute and the prisoner's ' history as a mediator. Details are in Supplement H.2
6.2 RESULTS
UDE
Results for the synthetic dataset: We show that our framework generates valid bounds (Fig, 4. Specifically, we demonstrate that our bounds successfully capture the true path-specific causal fairness. We find: (1) Our bounds (red) include the oracle effect as long as the sensitivity parameter is sufficiently large. This demonstrates the validity of our bounds. (2) The naïve effects, which assume no unobserved confounding (blue), differ from the oracle effects. This shows that failing to account for unobserved confounding will give wrong estimates of CF E {DE, IE, SE}.
UIB
Figure 5: Effectiveness of achieving causal fairness as measured
by DE, SF, (from left to right), which should be close to zero
We now compare our framework against the baselines (Fig. 5) based on the estimated path specisp causal fairness. We find: (1) The standard model (gray) fails to achieve causal fairness. The path- specific causal fairness effects clearly differ from zero
(2) The fair naïve model (purple) is better than the standard model but also frequently has path-specific causal fairness effects different from zero. (3) Our fair robust model (orange) performs best. The predictions are consistently better in terms of causal faireess. Further, the results have only little variability across different runs, adding to the robustness of our framework (Table 2 Importantly, our framework is also highly effective in dealing with larger confounding levels. See Supplement forr further results.
We aim to demonstrate the real-world applicability of our framework. Since benchmarking is impossible for real-world data, seek to provide insights into how our framework operates in practice. In Fig. & we compare the predicted prison sentence from () the standard model and (ii) our fair robust model. The standard model tends assign much longer prison sentences to non-white (than to white) offenders, which is due to historical bias and thus deemed unfair. In contrast, our fair robust model assigns prison sentences of similar length.
Results for the real-world dataset:
SE
Standard
upper
0.14 :: 0.00
-0.14 :1: 0.00
-0.25
0.01 1: 0.00
Fair naiiee
upper
0.06
0.05
-0.02
0.00 4 000
0.05
0.00 + a.0
Fair (ouns)
upper
0.00 = 0.02
0.01
-002 0.02
-001 0.01
0.00 1 0.000
0.00 4 0.00
Figure 6: Predicted prison sentence length (months) per race group from the standard model (left) and the ro- bustly fair model (right).
Table Estimated bounds on the synthetic dataset "UDE" for confounding level & 2; mean and std. over 5 runs.
DISCUSSION
Applicability: We provide general framework. First, our framework is directly relevant for many settings practice (see Supplement (B.I). Second, it can be used with both discrete and continuous features (see Supplement]. Third, it is not only applicable to the specific notions above, but it can also be employed for other notions causal fairness (see Supplement FF)
Failing to account for unobserved confounding may lead to wrong conclusions about whether a prediction model fulfills causal fairness. First, our framework can be used to perform sensitivity analysis for causal faimess of existing datasets. This is often relevant as information about sensitive attributes (e.g., race) cannot be collected due to privacy laws. Second, our framework can be used to test how robust prediction models under causal faimess constraints to potential unobserved confounding. To this end, our work is of direct practical value for ensuring the fairness of predictions in high-stakes applications.
Conclusion:

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A
NOTATION
A
Set of nodes corresponding sensitive attributes
M
Set nodes corresponding to mediators Set of nodes corresponding to observed confounders Set of nodes corresponding unobserved confounders
Z
U
UDE; UIE USE
Unobserved confounders on the direct, indirect and spuri- effect, respectively
Y.Y
Outcome and predicted outcome
X'
Set of possible values random variable Probability distribution over random variable
P())
p()
Probability density function of continuous random vari- able
P
Family of probability distributions
c
Structural causal model
g
A causal graph
Ang(z.)
Ancestors of r; in G
Pag(z:)
Parents of æ; in G
Generalized marginal sensitivity model
I'w
Sensitivity parameter
K(S)
Class of SCMs C compatible with sensitivity model S Counterfactual direct effect of a; wrt. dg
DE
DE ao DE 00
Upper and lower bound on counterfactual direct effect Counterfactual indirect effect of a; dg
LE ap IE 1.
Upper and lower bound on counterfactual indirect effect Counterfactual spurious effect of a; wrt.
SE
SE+ ag.n; SE
Upper and lower bound counterfactual spurious effect
CF, CF+ CF
Causal fairness notion with upper and lower bound
D
Functional mapping density to scalar value
fo
Prediction model with parameters
g
Density estimator
Fairness constraint
+
Confounding level
N(H, Σ)
Gaussian distribution with mean f and covariance
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B DISCUSSION OF APPLICATIONS
B.1 EXAMPLES: CAUSAL FAIRNESS WITH UNOBSERVED CONFOUNDING
Prediction of credit default risk: When ap- plying for loan, banks commonly evaluate the applicant's credit default risk based on various factors, such as the applicant's credit history, occupation, and the presence of a guarantor. Following the goal of profit maximization and risk minimization, the bank will only grant the loan to solvent applicants, i.e., applicants with a low default risk.
Occupation, amount, guarantor
Immigrant
Housing. credit history
Defbult
worker
Various fair lending laws across the globe quire that the risk assessment must be fair and, to this end, should not be affected by sensitive information. An example could be whether or not the applicant is an immigrant worker. The example is shown in Figure 7 Therein, the default risk estimated based on information about the occupation and wage of the applicant, the size of the loan, the presence of guarantor, the housing situation of the applicant (since hypothecary credits can be based on the property). and credit history. fair and strategy-optimizing prediction model should (i) control for any direct, indirect, and spurious effect stemming from the fact of applicant being an immigrant worker and (ii) incorporate the potential effect the unobserved variable race on the immigration as well credit history. Of note, laws in various countries forbid the collection and storage of informa- tion about race so that such information is often missing, and, therefore, presents an unobserved confounder.
Race
Senetive attritne
Outoome
Obberved vetteblee
Unobsonied conbunders
Figure 7: Example: causal graph for predicting credit default risk.
Training a machine learning method for predicting new applicant's default risk without accounting for faimness constraints and potential unobserved confounding will likely result in biased and unfair predictions and, thus, biased loan allocation. In the stated example, immigrant workers might not receive loan, although they would constitute a good risk in an unbiased assessment, whereas na- tionals with a high default risk might falsely be granted loan. This not only reinforces the societal bias present in the data but also fails to optimize the bank's strategy in terms of profit maximization and risk minimization.
Prediction of acceptance of child nursery: Several works aim to leverage machine learn- ing models in public organizations. One exam- ple is the following. Due to high birth rate and a shortage of available places at childcare facilities, not all children can often be admit- ted to a nursery. The decision of which child to admit might depend the parents occupation, the number of children in the family, and the housing situation. The causal diagram is shown in Figure where we detail the mechanisms behind a recommendation for admission to the nursery. In the present example, we argue that the financial status of the family should only have a limited effect on the recommendation of admission. i.c., the prediction unfairness intro- duced by the financial status should be lower than a specific threshold. Furthermore, multi- ple unobserved factors typically influence the family's 's financial status and the observed variables. For example, the financial status and occupa- tion could both be influenced by the family background, potentially mediated through the parent's
Occupation children, family structure
Famill background
Housing situation
status
dation
Neightouur head
Sonsiive attribute
Outoome
Obsereed varibles Unobuarved conbunders
Figure 8: Example: causal graph for predicting acceptance to the nursery.
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education. The residential neighborhood is related to the family's financial status and the housing situation. Then, a good financial standing might indicate the family's 's ability to pay for personalized private childcare (e.g., nanny) and, therefore, could have certain effect the recommendation for admission to a nursery. However, a machine learning model trained solely on the observed data will inevitably provide biased and unfair recommendations.
The above examples show two important applications of causal fairness in practice where unob- served confounding is prevalent and could be addressed by our framework.
B.2 DISCUSSION ABOUT KNOWLEDGE THE STRENGTH OF UNOBSERVED CONFOUNDING
In practice, knowledge of the unobserved confounding (UC) strength is beneficial but not fully necessary. Importantly our framework is of direct help in practice when (i) the maximum UC strength is known even when (ii) unknown. We discuss both use cases below:
1. There are often good reasons why the UC strength is known or can be upper-bounded. Many faimess-relevant applications are rooted in social science, where there is a good understanding of potential causes for biases (e.g., around gender, age, etc.), Hence, it is often reasonable to assume that the UC strength in specific relationship with some other observed variable. For example, based on domain knowledge, one can often say that the UC strength is not as strong as (a multiple of) the observed variable (e.g.. Cinelli & Hazlett 2020 Franks al. 2019 Masten & Poirier 2020 Zhang al., 2016). In such cases, the UC strength w can be set by practitioners accordingly, and our framework can be applied discussed in the main paper
2. Even when the UC strength is unknown, our framework is of significant value in practice. this case, practitioners can employ our bounds to test the sensitivity of causal fairness notions with respect to unobserved confounding in the data. One way to do so is to use our framework to calculate bounds for increasing sensitivity parameters until the interval defined by the bounds contains certain value of interest, e.g., zero (indicating complete fairness). The minimal sensitivity parameter to achieve this goal corresponds to the level of unobserved confounding, which would be necessary consider the data fair. Hence, one can also view the sensitivity parameter as our uncertainty about the fairness of our prediction model. Smaller sensitivity parameters that do not achieve the goal still pro- vide information about the sign, i.e., direction, of the unfair effect Hsu & Small (2013). Generally, testing various sensitivity parameters a common technique in real-workee ap- plications of sensitivity analysis (e.g., Hsu & Small 2013 Jin et al.) 2023] Vander Weele & Ding 2017), 7). In sum, even when the UC strength is unknown, our framework can thus be of large practical value.
Furthermore, several approaches have been proposed to choose sensitivity parameters in practice. Ya-
We refer to, c.g., Jin al.) (2023); Kallus et al. 2019); Diaz van der Laan (2013; Imai mamoto (2013) for further discussions.
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C EXTENDED RELATED WORK
C.I FAIRNESS IN MACHINE LEARNING
In the following, we present an extended discussion of related works. First, we give brief taxonomy of algorithmic fairness. Then, we discuss related causal faimnsss notions and explain the difference between counterfactual and causal fairness. For the latter, we heavily rely on Plecko & Bareinboim 2022) and refer to their work for an in in-depth discussion of causal fairess analysis.
There are mainly two classes of fairness notions, namely, statistical notions and notions based on causal reasoning (Fawkes et al. (2022). Often, statistical fairness notions are incompatible with one another (Fawkes et al. 2022; Kilbertus et al., 2019, Kleinberg et al., 2017) and cannot provide intuitive results (Nabi & Shpitser 2018). Causality-based notions can help in overcoming such problems. Relatedly, fairness notions can be defined on a group or individual level, where the specific choice depends on practical considerations (Loftus et al. 2018). Our framework focuses on group-level fairness. Although developed for training fair prediction models, the results can also be used explain individual discrimination in a prediction model.
Algorithmic fairness:
At a technical level, faimness in predictions can be achieved at different steps: during pre proccsssss,g in-processing, or post-processing Since a prediction model trained data fair labels is not nec- essarily fair itself (Ashurst et al. 2022), pre-processing fairness notions do not always mitigate the unfairness present in the resulting prediction. Therefore, in our work, we seek to ensure fairness in the presence of unobserved confounding through an in-processing approach.
Causal vs. counterfactual fairness: Causal fairness measures quantify the association of sensi- tive attribute and the target variable through causal mechanisms in a structural causal model (SCM) (Plecko& & |Bareinboim] (022). Specifically, causal fairness notion must fulfill the three elementary structural fairness criteria: (I) the structural direct criterion, assessing if the target is a function of the sensitive attribute; (2) the structural indirect criterion, determining the presence of an effect of a mediator variable on the target which in turn is influenced by the sensitive attribute; and (3) the structural spurious criterion, evaluating the presence of confounder between the sensitive attribute and the target. Causal fairness notions can, therefore, differentiate between direct and indirect dis- crimination to mathematically account for legal definitions of fairness, such as disparate impact and disparate treatment, incorporate business necessities.
As a specific notion, counterfactual fairness (Kusner et al., 2017) enforces the outcome variable to be identical in both the current observation and a hypothetical counterfactual world in which the protected attribute has a different realization. It specifically focuses on the total variation in the outcome caused by change in the sensitive attribute and does not differentiate between multiple pathways of discrimination. Many extensions and methods for ensuring counterfactual fairness have
been introduced in the literature (e.g.. .hiappaa 2019 Kilbertus et al. 2017 Ma et al. 2023).
Several causal fairness measures are also based on counterfactuals (see Plecko & Bareinboim (2022) for taxonomy overview). so warn that the naming in the literature may be misleading. There- fore, we highlight the differences in the following. Importantly, counterfactual faimess and any fairness notion built upon it differs from counterfactual-base based causal fairness notions in three fun- damental aspects: (1) admissibility, (2) ancestral closure, and (3) identifiabilit. We discuss the aspects in the following:
1. Admissibility: Even if the counterfactual fairness metric evaluates to zero and thus implies that a model is fair, there is still no guarantee for both the direct and indirect association of the sensitive attribute and the target being zero, the effects might cancel out. To this end, counterfactual fairness is inadmissible w.r.t the structural criteria.
2. Ancestral closure: Ancestral closure requires all ancestors of the sensitive attribute to be observed. This implies that there cannot be any unobserved confounders on an association containing the sensitive attribute, which is unlikely to hold in practice. Causal fairness notions, on the other hand, allow for endogenous ancestors.
3. Identifiability:
Throughout our paper, we have dealt with unidentifiability of causal fairness due to unobserved confounding, arguing that the assumption of no unobserved confound- ing does not hold in practice. If the full SCM was known, the faimess notion would be identifiable. Counterfactual fairness, however, is never identifiable from observational data
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if mediators between the sensitive attribute and the target exist, even if the full SCM is specified.
In sum, both causal vs. counterfactual fairness are entirely different notions. In our work, we focus on causal fairness.
C.2 CAUSAL SENSITIVITY ANALYSIS
Causal sensitivity analysis has widely been employed for partial identification problems. By impos- ing assumptions the strength of unobserved confounding and, therefore, relaxing the assumption of no unobserved confounding, various causal effects can be bounded the presence unobserved confounding. The resulting bounds often prove that the causal quantities of interest cannot be explained away by unobserved confounding. Especially for real-world studies and policy learning, sensitivity models thus provide important tools for assessing the sensitivity of causal estimate to unobserved confounding through deriving informative regions for the causal effect (Kallus et al. 2019).
The main sensitivity models introduced in the literature are: the sensitivity model of Rosenbaum (Rosenbaum] 1987) employing randomization tests; the non-parametric marginal sensitivity model Tan 2006); and the recent f-sensiiity models (Jin et al. (2023). For binary treatments A E {0, 1),
the marginal sensitivity model (MSM) Tan (2006) defined as
(2)) 1 1 - u) ST,
(12)
r
1- TT2
where ((2)=
P(A= | denotes the observed propensity score and π(2, u) = P(4 = 1 2. 1
denotes the full propensity score.
Multiple extensions have been developed recently, especially for the MSM, which has received much attention in the literature. Most work has focused the average treatment effect (ATE) and
conditional average treatment effect (CATE) for binary treatment settings (e.g., Dorn & Guo 2022
Dorn et al. 2022 Jesson et al., 2021 Kallus et al.) 2019: Oprescu et al., 2023 7Zhao et al.] 2019)
Popular extensions for the MSM include sensitivity models for continuous treatments (Jesson et al.
2022 Marmarelis al. 2023a). For continuous treatments A, the continuous marginal sensitivity
model (CMSM) Jesson et al. (2022 is defined through
1 P(a 2,
S
ST.
(13)
F
1)
Other extensions include augmentations to individual treatment effects (Yin et al. 2022 Jin et al. 2023 Marmarelis et al. 2023b).
The work by Frauen et al. (023b) introduces the Generalized Marginal Sensitivity Model (GMSM): For an observational distribution Py on V and a family P of joint probability distributions on VUU that satisfy
(1 - (w-Tw) qw (a, 2) +Iw S P(U uw 22
5 r 9w (a 2) +
(14)
P(U = uw z, 2,d)(a))
for W E {M,Y] and weight function (a z) E [0, 1]. the tuple S = (V, U.G, Pv, P) is called weighted generalized marginal sensitivity model (GMSM) with sensitivity parameter Iw 1. can be shown that, for weight functions (a. 2) P(a 2) (MSM) and q(a. 2) = 0, the MSM and the CMSM are special cases of the weighted GMSM (Frauen al. 20236). Therefore, the GMSM generalizes many of the aforementioned approaches it allows to derive bounds for binary and continuous treatments as well as different causal queries (e.g.. CATE, distributional effects).
Causal sensitivity analysis has also been applied to other settings, such as off-policy learning (e.g.. Halt et al. 2022 Kallus Zhou (2018) or partial identification of counterfactual queries (Melnychuk) al. 2023). Recently, neural frameworks for automated generalized sensitivity analysis have been proposed (Frauen et al. 2024). Only one other work besides us has employed sensitivity analysis to study fairesss notions under unobserved confounding (Kilbertus et al. 2019). Nevertheless, this work focuses on the notion counterfactual fairness (and not causal fairess) and is limited to non-linear additive noise models.
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Rationale behind our choice of sensitivity models:
In the following, we compare multiple extensions of the MSM with respect to applicability and discuss the strengths and weaknesses. Thereby, provide justification of why we adopt the GMSM in our paper.
A general benefit of the MSM is that it does not impose parametric assumptions the data- generating process, thus enabling wide applicability to many domains. Nevertheless, the sensitivity model is only defined for single binary treatment variable. Mutliple extensions derive bounds based on the MSM (e.g.. Dorn & Guo 2022 Dorn et al. 2022 Jesson et al., 2021 Kallus et al.)
2019 Oprescu et al. 2023] Zhao al. 20T9). However, the bounds provided by the extensions
above are unnecessarily conservative. Therefore, Dorn & Guo (2022) and Jin et al. (023) derived closed-form solutions for sharp bounds under the MSM.
The main weaknesses of the above extensions are (i) the limitation to binary treatment and (ii) the restricted focus on the (conditional) average treatment effect. To overcome weakness (i), Bonvini et al. 2022);; Jesson et al. (2022) and Marmarelis et al. (2023a) proposed extensions for continuous treatments. Additionally, Bonvini et al. (2022) extended the setting to time-varying treatment and confounding variables. To overcome weakness (ii), Jin et al. 2023); Marmarelis et al. (2023b) and Yin et al. 2022) introduced sensitivity models to cover the individual treatment effect (ITE).
In our work, we adopt the GMSM. The generalized marginal sensitivity model (GMSM) (Frauen ct al. 2023b) provides a general causal sensitivity framework that can incorporate continuous, dis- crete, and time-varying treatments. The resulting bounds are sharp and applicable to multiple causal effects (e.g., CATE, ATE) as well as to mediation analysis, path analysis, and for distributional ef- fects. Overall, the GMSM is thus highly suitable for deriving bounds causal fairness metrics and thus makes our framework widely applicable to various settings (e.g.. discrete and continuous variables, etc.). Of note, we the GMSM only formalize our setting, while the actual derivation of bounds is non-trivial (and is not direction application of the GMSM). The reason is that the GMSM makes interventional queries (=level in Pearl's causality ladder), while our task involves counterfactual queries (=level 3). Hence, existing bounds are not applicable; instead, new and careful derivation of bounds that are tailored to our setting is needed.
We also summarize the above discussion in Table which provides systematic overview of the applicability of existing MSM extensions.
Table Overview key extensions of the MSM causal sensitivity analyses. Applicability'non- applicability is indicated by green tick (V) and red cross (X), respectively.
MSM exensioos
Treatment type
Causal query
Binary
(CATE
Distrilutional effects
Individual effect (TE
Tun 200S
2019
x
x
Xxm & Lo
x
LXT 21 11
/
X
2033
X
X
X
Teoos 0 31
2144
X
/
X
X
2023g
7
X
Yin e
2200
X
X
20236
X
X
C.3 CHALLENGES IN SENSITIVITY ANALYSIS FOR CAUSAL FAIRNESS
We transfer concepts from sensitivity analysis to the causal fairness literature by (i) showing the applicability of sensitivity analysis outside the field of standard causal effects, and (ii) providing a solution tool for assessing causal fairness under unobserved confounding well direction for future research. Nevertheless, sensitivity models are not directly compatible with causal fairness notions. We elaborate the difficulty and proposed solution below.
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In the following, rephrase workflow in terms of Pearl's ladder of causality (Fig. We aim to provide bounds for causal faimess notions, which can be estimated from the data. Causal fairness notions are located level three of Pearl's ladder, i.c., they contain counterfactual expressions. However, sensitivity models are interventional queries and thus located on level two. Hence, existing bounds from sensitivity models framework to bridge the gap between levels three and two.
applicable. To remedy the above, need to develop
Therefore, we propose the following approach to deriving bounds: we propose decompose the nested counterfactuals into interventional (non-identifiable due to unobserved confounding) and identifiable effects. This step is non-trivial and requires customization for every causal fairness notion. Then, (ii) we employ sensitivity analysis to derive bounds on the interventional terms. The resulting fairness bounds consist of concatenation of sensitivity bounds and the decomposed identifiable effects. We present a schematic of our workflow in terms of the ladder of causality in Fig. 9
3: Counterfactual
Causal fairness (null-rderventoo)) on counterfnctutts
2: Intervention
Unnested CF single intervnntinss
Counterfactual unnesting identification
Association
Fairness bounds flation
Sensitivity analysis
Figure 9: Our workflow in terms of Pearl's ladder of causality. Note: sensitivity models like the GMSM are interventional queries (=level 2 in Pearl's causality ladder), while our task involves counterfactual queries (=level 3), because of which a tailored framework for deriving bounds is needed.
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D PROOFS
In the following, we derive proofs for our main theorem from Section According to the three-step approach presented in Section we first provide theoretical background on the sensitivity model S (the GMSM). Then, we perform counterfactual unnesting of the path-specific causal effects. Finally, prove the overall bounds per counterfactual effect.
D.1 THEORETICAL BACKGROUND ON GMSMS
We can derive bounds ligle-coouteerfcuue causal effects, i.e. unnested counterfactuals, in GMSM based on the following theorem.
Theorem 2. (Frauen al. 2023b) Let S be a GMSM with sensitivity parameters Iww, W E MU }. Further, we restrict each unobserved confounder U E U to be the parent of only one element in M U {Y), i.e., there does exist unobserved confounding between mediators and outcome as well as between mediators themselves. Let K((S) denote the class of SCMs C compatible with S. Let F.) as denote the CDF corresponding to P- Z, Mw. A). For a continuous W, we define
P(w 2, mww a),
if F(w) S
Tw
P+ 2, k.W, a)
T+TW Tw
(15)
P(w z, w,w, 0),
if FfW(w) >
I4TW
and
1 P(w mwwa), a), ifF(w) S ItIW
(- z, z,
(16)
P(w 2, k,,,aa) a), if F(w) I+IW
s
For discrete W N. we define
P(w] m, a)
if <
Tw
I+TW
pt(w|] 2, (.mmaa a)=
P(w 2, k, a);
if I+IW
Twr
Tw F(w - 1)) sw I+I'
(F(w)
Tw I4f
)
otherwise.
(17)
and
P(w 2, m.w a):
F < TTTW if ([ww1) I+TW
W P(w 2, mW a))
P- (w) 2, m, a)
W
F(w 1)) _
(F(ww) I+T
otherwise.
I+TW
(18)
With (+),,, F- (-) and Fe(-) denoting the conditional CDFs corresponding to P+ 2,m,,, a). P- (w m. a) and Pel- z, mw do(A = a)). respectively, we yield and
F+()) inf Fe(w) CEK(S)
F F- sup Fc(w)
(19)
CEK(S)
for all w.
D.2 COUNTERFACTUAL UNNESTING
We perform counterfactual unnesting based on the counterfactual identifiability theory presented in (Correa et al., 202T). We adopt the notation therein. Let Vx, V E V.X C denote counterfactual expression for V. The ancestral set of Vx given by Anx(Vx) (W, W Angy (V), Xn Angy (V)). where Gx denotes the graph deduced from & when removing all edges going out of variables X.
For the general case, let W, denote set of arbitrary counterfactual variables. The corresponding ancestral set is defined as Ang (W.) = Uwiew, vention.
where t denotes the respective inter-
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Theorem (Ancestral set factorization from (Correa al. Ang = W., and let ws, be realization of wx. Then,
2021 ). Let W, be ancestral set, i.e.,
P(W. - -..)= P
A WiEW.
W Wor(W)+ = w/
(20)
where the subscript denotes the set of interventions W present in W.. Note that might be the empty set, i.e. the variable W is not intervened on. The set of interventions pa(W)) for each W, W. corresponds to the interventions for non-empty Otherwise, pa(W) is given by the parents (W) of W.
Analyzing counterfactuals such as in Eq. 20 based on their counterfactual factors ('etf-factors') can aid in deciding its identifiability in a given graph. A counterfactual factor P(W, | wi) for W; V, where potentially W; W for some i,j [I.....I generalizes the parent-child relationship encoded in e-factors to the counterfactual domain. The brackets around the subscript denote the interventions on the enumerated variables W; for E (I.....I).
W1, W2 (m(t))) W22, .... jeccn)
Theorem (Counterfactual factorization from (Correa al., 2021 )). Let P(W. w.) be coun- terfactual factor with a topological order over the variables in G G[V(W,)] Further, let C.....,C Ck be the e-components of the stated graph. Cj+ = (W)) W, W C5} with cj the values in
W.. Then, one yields
P(W. W.) II P PC,, =c,.).
(21)
In the following, we will use the stated counterfactual unnesting theory to split the expressions P(ya, a) and P(yar.ma a) into the respective counterfactual factors. We thus obtain
P(ya, aa)
P(y., Ma,)
p, ags Ma..)s
(22)
P(a)
P(a)
=EZ.
ZEZ.
MEM
mEM
where we leverage the ancestral set expansion (*) and the counterfactual factorization (**) from above, respectively. If we assume there exists unobserved confounding between sensitive attribute and mediator (UIE), sensitive attribute and covariates (USE), and sensitive attribute and outcome (UDE), this expression cannot be reduced further.
The expression for the second term of interest follows similarly via
P(ya,oma maj a0)
Σ MEM
po a, May)
Σ P(ya,eemm Ogs 2, Majie).
(23)
P(a)
P(a)
ZEZ.
ME
As result, write path-specific causal faimess combination of identifiable effects and single-counterfactual effects, which subsequently can bounded following Theorem Hence, we yield
DE a (y as) = =P(ya, aas) P(ya; |aa)
(24)
P(yo az, 2, May+)) P(y as)
(25)
P(a:)
ZEZ.
mEM
P(u|,, dola))) d)(a;) (
(26)
P(as)
=EZ.
P(ay)
Σ
Ply m, m,2,a,))((m,. a:) - Ply as).
P(a:)
ZEZ.
mEM
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aE (y 1ylly aa) = =Payas,ma, aa,) - P(yas 1 lay)
(27)
P O2; 2,Mazz,)
Σ
2 Oz F, max
(28)
P(aj)
sEZ.
=EZ.
MEM
mEM
P(aj) Σ
P(ulz, m, do(a;)) 2)(a)),
(29)
sEZ,
mEM
P(a:)
P(y m,, |a))
P(ay)
EEZ.
(P(y, z, do(a;)) P(y, 2, P(a)))
P(a))
MEM
SEara, a, (1) - =P(ya: |0,)
Ply az)
(30)
P(yoramm 0z2 R, Ma,;)) Ply a.)
(31)
P(ay)
=EZ. mEM
Σ
(P(y,2, do(a;)) P(y,z,m (1),,,/la))((a) a():) P(y aa))
P(as)
MEM
In the above equations, the effects with do-interventions are unidentifiable in the presence of unob- served confounding. Furthermore, present details on the performed calculations in the proofs of LemmaAland and Lemma 3
D.3 COUNTERFACTUAL IDENTIFICATION AND BOUNDS
We use the following corollary.
Corollary 1. Let X and A be two variables of structural causal model. By consistency it holds
P(X.) a)
+ P(Xa
P(A
(32)
Proof. Follows directly from basic probability theory.
We start by deriving bounds for expressions of the form P(ya. a). If as there is nothing to show, since, by consistency, we have P(yo; a.) P( a.). Therefore, let a + a, in the following.
Lemma 2. For a binary sensitive attribute with a, # aj. holds
p+(y z, m, a,m a, a, z)(2) - P(a+) P(aj)
p+ (Ya, |a,) P(az)
Σ
Σ
P(y. |a)),
ZEZ. MEM
=EZ.
mcM
(33)
P(a:)
P | = P(a;)
Σ P- 2 P-m. a, P..2)((2))
Σ
P(y. z,m a.).
P(aj)
ZEZ
sEZ.
MEM
mcM
(34)
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Proof. By Theorem
we can write
P(ya; aj) Σ
P(ya;: 2, maz la²)
=EZ. mEM
1
P(y,
do(a;)) ( P(u, 2, la))
P(aj)
(35)
sEZ, mEM
E
2 ma, dd)P(a'))
t000 a' Fay
For binary sensitive attributes, this reduces
P(yas (a))
[P(y, do(a.)) Ply, 2, Pss))]
(36)
P(aj)
SEZ.
TEM
The first term is not identifiable but can be bounded in GMSM by rewriting the term
Ply,2,m do(a:))
Σ
Ply z, (( do((a))))P((
(37)
MEM
MEM
Then, the desired statement follows by Theorem
With the results from above, we turn to the derivation of bounds for the second factor P P(ya, la). Based on the causal explanation formula, we only have to consider the case a aj.
Lemma 3. For binary sensitive attributes, it holds
P* 1a,) = P(ay)
Σ P ( ag, P(2)
2EZ.
MEM
(38)
P(as)
P(aj)
ZEZ.
MEM
p- (ya
(aj) P(a;)
1
Σ l a;) (m la;;))(;
sEZ.
mEM
Σ Plylm,z,a:)PPpl ( (39)
P(a.)
P(aj)
ZEZ.
MEM
Proof. According to Section D.2
we can write
1
P(yarma, |a,) = P(a,)
P(ym ((m ((22
ZEZ. mEM
P(a:)
Σ
Ply m, 2,2,a.) a:)
P(ay)
(40)
ZeZ.
mEM
Σ Σ
P(ya..m Ma, P(a))
P(aj)
ZEZ,
1iij
MEM
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If A is binary, this reduces
P(ya,,aa
|ay)
Σ
P(y 2, m, p.m.ddd)))((mmdddd.)).))))))))
P(aj)
=EZ,
mEM
(41)
Σ Plylm,,,a))
Σ
Plyms May1 x os).
P(aj)
mEM
mEM
We further use
1 Ply may 2, do(a:)) 2, 2,do(a))))((2)
m Mays 2 |ai) P(a:)
(42)
P((a)) P(um, May do(ai); =P(a,)
and, then, the overall expression follows, i.e.,
P(ya,me, |,,) = P(az)
m, do(aa)))mmm =)
ZEZ
mEM
(43)
P(a:)
Σ
P(y m, 2, 2,2,a)) la)).
P(a))
SEZ,
mEM
Overall, we want to bound the path-specific causal fairness effects (i.e., Ctf-DE, Ctf-IE, and Ctf-SE), which consist of differences of the two expressions analyzed above. In the following, we derive the bounds for each counterfactual factor.
Proof for Theorem 1
Employing Theorem
Lemma and Lemma 3. the counterfactual direct effect is upper bounded
by
DE ag. (y a.)) = P (Vay,ma; aa,) - P
PP (ya: aas)
(44)
1
p+ m, at) a, a,,z) P())
(45)
P(a:)
EE
MEM
P(aj)
Σ
P(y m. (m22,,,)((m22,..)
Ply as)
P(as)
=EZ.
MEM
and lower bounded by
DE (y a;) P (Yo, ma:
(ai) p
(Ma, PT
aa))
(46)
Σ p- m, 2, a.) (m 22,
(47)
P(as)
=EZ,
mEM
P(ag)
Σ
P(y m. z, a,),, 2, 2(())((2) - P(y a:).
P(a.)
2eZ
meM
The results for the indirect counterfactual effect follow directly by setting
IE+ (y aa) pt (Ya,,a, |a,) - p P- a.)
(48)
and
T (y (a)) P (Yagmm. (as) p (ya, az)) Similarly, the bounds for the spurious counterfactual effect can be derived
(49)
SE Bo.ay (v) p (Va, aa)) - P(y a;)
(50)
and
SE SE a, (v) P (Ya. aa,) Ply a.).
(51)
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E EXTENDED THEORY
E.1 COMPATIBILITY OF SCMs
acyclic SCM C (V. Uc. Pu), such that U Uc. - Ge. compatible with sensitivity model S if Uc does not introduce additional unobserved confounding and the probability distribu- tion PvLue induced by C belongs to P, i.e., Prut, P.P
We further assume additional unobserved confounding. That is, we have: For each M denote (M)(M) the observed parents in V. Then, a valid adjustment set for the relationship between Pag(M) and M, i... P(m| do(Pag = pa)) = P(m do(Pag(M) pa), U = P(U u)du.
E.2 EXTENSION OF THEOREM
In the following, we provide an extension of Theorem for bounding the counterfactual effects of continuous outcome variables.
Lemma 4. Let Fly) denote the CDF of Pyy m, a.) for continuous outcome Y. Then, the probability functions p* (y m, a;) and p (y m, (s) are given by
((1 - (y-- P(a. + 1o++T')pps m,),a,), FF(()) s 1 (52) F( > 1
Tv
p² m, ai)
((1 + +Tr)p(y m,
and
((1 [[1-Ty)P(aa +T lply m, m, ((1 [ry)- 2) + (eTr))(l m, 2,)),
F I I4fY if F(y)) > 14
pr (3) m,
(53)
The bounds require estimating the observational density ply 2, m, a). Nevertheless, in practice, can only observe the empirical distribution from which we can obtain p(y x, m, a) via an arbitrary conditional density estimator. The work in (Frauen al. (023b) introduces importance sampling estimators for finite sample hounds, which can be adapted to estimate the observational density.
E.3 COUNTERFACTUAL IDENTIFIABILITY FOR -UB-PROBLEMS
Our work obtains bounds the counterfactual effects in general setting with unobserved founding, i.e., U = (UDE, ULE USE} * 0. Now, we state the counterfactual identification results for subproblems in which subsets of U considered to be zero.
In Section D we employ counterfactual unnesting theory to split the expressions P(ya, | a) and P Plya,,ma a) into the respective counterfactual factors. For the first factor, we derive
P(yap
aa)
P(a) P(yaao.amm ags Z, Mag.s)
(54)
If assume that there exists unobserved confounding between sensitive attribute and mediator (UrE), sensitive attribute and covariates (UsE), and sensitive attribute and outcome (UDE), this ex- pression cannot be reduced further. Other possible scenarios of unobserved confounding exam-
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ined in the following. By the counterfactual factorization theorem, yield
UDE UiE = USB = 0:
P(yag a) =
Σ P(yav. ( P(a)))(())P(mmmm)
(55)
2,m
USE + 0, UDE = UiE 0:
P(yan a)
=
Σ P(yao.z. P(az, 2) P(man:)):
(56)
UIE * 0, UDE
USE 0:
P(yag+ a)
Σ P(yav.z. P(az, mag P(z),
(57)
UDE + 0, USE Uip 0:
P(yag a)
Σ Plyao.z.ms ((m
(58)
USE, UIE * 0, UDE 0: P(yag a)
Σ P P 2, mao.))
(59)
2,973
USE, UDE + 0, UIE O
P(yag: a)
Σ P(yao.z.m Oz 2) P(Moo,
(60)
2,372
UIE UDE * 0, USE = 0
P(yago a)
Σ
P(yao... ag Mag P(2).
(61)
2,77
All terms in the first two cases are directly identifiable from the data. In all other cases, one term is unidentifiable for & ag- To derive our bounds, we focus on the general case, allowing for all three types of unobserved confounding. The expression for the second term of interest follows similarly.
E.4 TRAINING ALGORITHM FOR FAIR PREDICTION
In the following, discuss Algorithm| in more detail. Depending the task, i.e., binary clas- sification, multi-class classification, or regression, the precise calculations of the constraints might vary. Therefore, we will discuss all tasks separately:
1. Binary classification: Assume A. E 1). For each of the three different fairness effects CF E {DE, IE, SE), we are interested in, e.g., CF((Y = 1 A = 1). Due to sym- metry reasons for binary sensitive attributes A (e.g.. discrimination of women compared to men symmetric to discrimination of men compared women), it is sufficient to focus realization of the sensitive attribute. Therefore, obtain only constraint per CF E (DE, IE, SE}, i.e., three constraints in total.
2. Regression: For continuous outcome variables Y E R, reasonable to consider the expectation over Y. Therefore, for this task, we also obtain one constraint per CF and thus three constraints overall.
3. Multi-class classification: For this task, two different options exist for defining the con- straints aim to optimize over. Option one follows the same reasoning in binary classification and regression in that one can constrain the expected CF over all realizations of Y for each CF E (DE, IF, SF). A more fine-grained approach is constrain each CF for each possible realization of Y. E y. As a result, we need to optimize our predictor | constraints. Although this feasible for binary classification, with respect challenging for large [)'). Which of these options to choose for a specific can be highly problem left to the practitioner.
Algorithm (main paper) provides the training procedure for binary classification and regression. In Algorithm 2 we present an alternative training procedure for multi-class classification in which impose constraint for each CF E (DE, TE, y E V.
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Algorithm 2: Alternative training procedure for fair multi-class classification models robust to unobserved confounding
Imput: Data [[A,, ZG. Mr, 2 (1,...; (.). sansitivity parameter I' 'M. vector tainess constraints 7y Lagrangian parameter voctors
An and Ma- pre-trainee densiry estimatoos 9.4 9M inital prediction model fay convergence enterion e, Lagranguan update rale a
Outpot: Fair prodictions (9, E (l.... x)) trained prediction model /
7+ Initialia
foy Sop AA prodiction model
:M JAg mux ftentions
nested spectss
9 +- (s (A, 2. Mr);
Determine CF C CP for CF (DE, IE, SE) following Theozen Foz each 3
CF - ublfo (A, M,), AM. v). Vy E 3%
CR - Iblfo, (4, Mar
1),
Vy
optimization objective zollowing Be. (DE, SR),
31
<CFy - max ((CF5 (CFyT])
end
c (EDRy1 ODE 918 "1 98 .g T.
where 13b
10
lagrangian + lo,) Au ape
(Au A
Update paraneters predictor
11
f + optimlser(lagraggiaa fo,)
12
end
13
if then
convergence
14
prodiction fo, ) S then
15
10
stu
17
end
Update tegrangian paraneters Ax+1 max(Ag oH,, MM ope.
19
20 end
We note that, depending on the task, the algorithms optimize with respect to three or | con- straints. Therefore, the vectors H, J. and the final fairness vector also of these dimensions.
We also note the following: In Algorithm (main paper), the maximum in Line always taken over the two values [E(CF+) and JE (CF ) for each CF {DE, IE, SE). In Algorithm 2 the maximum in Line 7 is computed over the two values |CF and CCF, y for each CF x y E D
The two algorithms thus differ concerning the number of evaluations of the maximum, ie.. three evaluations in Alg. (main paper) and x [)] in Alg. 2
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F BOUNDS ON FURTHER FAIRNESS METRICS
Here, show that our framework general and can be applied broad set of fairness notions. Our main paper focused on faimess bounds based on the path-specific causal faimess effects ZZangg & Bareinboim 2018a). Nevertheless, our approach general and can be easily applied other notions of faimess as well. In the following, we outline the derivation of bounds for the fair on average causal effect (Khademi et al. 2019) and path-specific individual fairness (Chiappa 2019 Wu et al. 2019b).
F.1 EXTENSION TO THE FAIR ON AVERAGE CAUSAL EFFECT (FACE)
Khademi et al. 2019) introduced the definition of a fair average causal effect (FACE) of group fairness, which we denote by FACE((;)). We further define the generalized form of the effect for non-binary attributes average FACE
AFACE(A) == [A = EA Y do(A = a;)]] - Ey do(A = -oo)]-
(62)
fair predictor should ideally return FaCE(a;) 0 for all a, E A. which implies AFACE(A)
O. Nevertheless, the single effects might not always be identifiable. In these cases, the average FACE can be used relaxation.
For j 1,....... h = |A], we define
EM Σ Σ ZEZ MEM
BI 2, (o 2, (())
The upper and lower bounds for B[ý 2, 2, do(a;)) and P(m z, do(o)) iteratively be derived through the algorithm for causal sensitivity analysis with mediators presented in (Frauen) al. 2023b). The upper bound uha, and lower bound lbu, of FACE(a,) (with respect baseline ao) are then given by
uba;
P())
Σ
|
a,m, 2, E (Y a. (m 2, 2,0;))..
sEZ
mEM
(63)
Ibos
EP()) Σ [E ( am (m 2,0;)) E (Y 2, maj) Pt (m 2, az) (64)
ZEZ
meM
For the bounds for AFACE(A), |A|= k, it follows that
ub
Σ wbas
lb
Σ
lbat
(65)
k-1
k
d'EAi0
a'EAiao
We note that the above bounds should be used cautiously since highly positive and negative effects can cancel out in the definition for AFACE. However, this is not due to our derivation but due to the definition of considering average effects at the population level without analyzing the heterogeneity across the population.
F.2 EXTENSION TO PATH- SPECICECIIIC INDIVIDUAL FAIRNESS
Path-specific individual fairesss (Chiappa 2019 Wu et al. 2019b along the path æ wrt. sensitive attributes an, aj is achieved, for all Z.
By Yo, [Ya Y.. 2 2|
(66)
holds. To build fair prediction framework based on this fairness notion, we need to derive bounds on the expressions P(Ya11x x 7 = 2) and P(Ya, 7 z).
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For simplicity, assume wlog. that represents the path A + M Y for single mediator M. Then, for the first term, it holds that
1/ = y Z = 2)
(67)
Σ P(Y,) y, M maa z ==)) 2)
(68)
Σ P(Y = y do(A a1); M m, 7 P( m do(A a1), 2 2).
(69)
The second term directly decomposes into
P(Y.. Z = 2)
(70)
Σ P(Y = y do A ao), M = m, Z = PM = m do(A = 4n), 2 = 2). =)) ME.
(71)
We observe that both terms constitute subproblems of our bounds derived in Supplement D The terms do not contain nested counterfactual expressions but only interventional effects. Therefore, can obtain the bounds from Theorem 2
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G IMPLEMENTATION DETAILS
We implemented the experiments in PyTorch Lightning. Code and data for the reproducibility of the results are provided in our GitHub repository.
All neural network classifiers used in our study consisted of three layers with leaky ReLU activation function and dropout. For optimization, used Adam (Kingma & Ba 2015).
Our proposed fair predictor training framework requires a pre-trained model for estimating the den- sity of the mediators based the confounders and the sensitive attribute. The density of the sensitive attribute conditioned on the confounders is taken as the relative frequency. Since we only consider discrete mediators, simple multi-class classifier is sufficient to perform the density estimation. We use the same neural network architecture for mediator prediction as for our standard classifier.
Hyperparameters of the classification models were kept fixed across the three classifiers on the simulated datasets for better comparability. The models consisted of one hidden layer of size ten and dropout layer with a rate of 0.1 and were trained with a batch size of 128 and an initial learning rate of 0.0001. We trained the fair naïve model and the fair robust model with faimesss constraint of = 0.02, initial Lagrangian parameters = 0.1, μ = 0.02, and an update rate a = 1.5. The prediction performance presented in terms of the ROC AUC is presented in Table To facilitate the evaluation of the overall performance (i.e. performance and fairness performance), we introduce a faimess utility function.
Definition 3 (Fairness utility). We define the utility of a fair predictor as the weighted sum of its fairness and its prediction performance measured as the ROC AUC: For w E [0
u(fe) wR(fe) (1- F)
(72)
where F represents the faimeess evaluated through causal fairness notion and R the ROC AUC.
We set ((fe) 1 ( DEg max lEgl. IEg |} + max [SE (}) and w = 0.5 for our calculations. We report the performance terms of the fairness utilities of all classifiers as well in Table Note that the defined utility is upper bounded by one but does not have lower bound. Thus, negative utilities are possible.
Table 4: Performance of the prediction models on the simulated datasets measured by the ROC AUC and the fairness utility.
ROC AUC
Faimess utility
Experiment
Standard
Fair nalve
Fair robust
Standard
Fair naive
Fair robust
PUDE
0.8579
0.8112
0.6780
0.7284
0.8074
0.5004
PUpe = 2
0.8245
0.7885
0.7618
00.7402
0.8138
00.6622
PUDE = 3
0,8037
0,8003
0.5226
0..550
0.8137
-0.5871
PUDE 0.7535
0.7466
0.5754
0.7809
0.8409
00.5711
UhE 1
0.8929
0.8671
0.7187
0.9026
0.6405
0.5334
Uhe 2
0.9150
0.8667
0,6856
0.8948
0.6348
0..332
PUE
0.8964
0.8271
0.7382
-0.9066
0.6497
-0.5145
D 4
0.8557
0.8254
0.7321
-0.9276
0.6581
-0.5200
For our real-world case study, we also employed three-layered networks with leaky ReLU activation function and dropout. The robustly fair model was trained with fairness constraint of = 0, initial Lagrangian parameters
3.0, μ 1.5, an update rate of 1.5, and sensitivity parameter of I 2. The hyperparameters (hidden layer dimension, dropout rate, learning rate) of the density estimator and prediction model were optimized with the library Optuna (https:// loptuna. org/) across 100 trials. The final parameters and the prediction performance measured by the MSE are stated in T
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Table 5: Hyperparameters and prediction performance of the mediator density estimator and predic- tion model on the real-world dataset.
Model
MSE
Learning Hidden dimension
Dropout
Density estimator
0.0009
256
0.2100
Standard predictor
3.2383
0.0294
32
0.3175
Robustly fair predictor
4.9607
0.0001
0.4503
H EVALUATION SETTINGS
H.1 SYNTHETIC DATASET
We consider two settings with different types of unobserved confounding. Both contain a single bi- nary sensitive attribute, mediator, confounder, and outcome variable. We call the first setting "UDE". Therein, we introduce an unobserved confounder on the direct effect with confounding level t. We call the second setting "UIF". Therein, introduce an unobserved confounder the indirect effect with confounding level Φ.
We then generate synthetic datasets for each setting and confounding level. Specifically, draw each 20,000 samples from the following structural equations for UD, UIE ~ N(2, e with & e [1, 4]:
Z ~ Bernoulli(0.5)
A Bernoulli(a(5Z UDE))
M ~ Bernoulli(o(44 + 27)
Y Bernoulli(r + + 2M UDE)),
and
Z (ernoulli(055)
A ~ Bernouli(o(5Z - UIE)
Bernoulli(o[AA)) 22 U1E)
Y (3r1 + Z 2M)),
where a is the sigmoid function. We restrict the Bemoulli probability to fall in the interval [0.02, 0.98) guarantee overlap. The simulation is performed through the partially randomized
causal simulator PARCS (attps: /ttt:a////ppp...ccr//ooeeee///ss
H.2 REAL-WORLD DATASET
Our real-world study is based on the U.S. Survey of Prison Inmates (SPI); see United States. Bureau of Justice Statistics (2021). Based on the survey data, we created dataset for predicting prison sentences for drug offenders. We consider the race of the defendant as the sensitive attribute and prison history mediating factor. Overall, the prison sentence should only depend the type of offense. As unobserved confounders, we consider the defendant's family's 's crime history and citizenship. Formally, filter the survey data based the following criteria:
General filter criterion "drug offense": Our analysis only considers prisoners sentenced for drug offenses. Therefore, we filter the complete survey data for offense type "drug" Furthermore, we split the type into "trafficking" and "possession", represented through binary variable, and disregard other non-specified drug offenses.
Race: We aim diminish the effects of race, especially "white American". on the prison sentence. Therefore, we combine all information about the defendant's race into binary variable representing' "white" and "non-white, respectively.
Offense type: We encode the offense type into multiple binary variables indicating drug possession, manufacturing or growing, import of drugs to the US, distribution of drugs, and drug money laundry.
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We use information about the prison sentence length in months continuous target variable. We restrict the imprisonment length to a maximum of one year for our analysis.
Sentence:
Crime history: Prior court verdicts can characterize crime history. We encode information about the defendant's 's prior sentences into discrete variable indicating the severity of the crime history, if any.
We are interested if family member of the defendant had been sentenced to prison before. Therefore, we combine all survey information about the former impris- onment of parents, children, and spouses into a binary variable indicating if at least one family member had been imprisoned before.
Family history:
Citizenship Immigrant the U.S.: The defendant's citizenship (if it is non-U.S.) is not included in the publicly available data due to privacy concerns. Hence, we use information about U.S. citizenship proxy, which we encode as binary variable.
Individuals who did not provide answer the respective question who answered "I do not know" were excluded from our analysis. We imputed missing target values through ANN-imputation with 10 neighbors. We split the resulting dataset in the ratio 60%, 20% 20% for training, validating. and testing our models, respectively.
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1 ADDITIONAL EXPERIMENTAL RESULTS
In the following, we present further experimental results to demonstrate our framework.
I.1 BOUNDS FOR VARYING SENSITIVITY PARAMETERS
In Figure and Figure [1] we present results for the fairnsss of the predictions. Here, we examine the three classifiers as in our main paper but when using varying sensitivity parameters variable in {1.2, 5.0) for calculating the bounds. We observe the same behavior as in our main paper: the interval defined through the bounds increases in the sensitivity parameter. For the standard prediction model, the interval eventually covers the original path-specific effect in the data. Together, the results further demonstrate the effectiveness of our framework.
carlaniing
Figure 10: Bounds of the three prediction models on the dataset including direct unobserved con- founding on DE, IE, SE (left to right) for increasing sensitivity parameter (top to bottom).
Figure 11: Bounds of the three prediction models on the dataset including indirect unobserved confounding on DE, IE, SE (left to right) for increasing sensitivity parameter (top to bottom).
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1.2 PERFORMANCE COMPARISON FOR VARYING SENSITIVITY PARAMETER
We now assess the robustness of our frame- work with respect the sensitivity parame- ter used for training the fair prediction model, ie., the level of unobserved confounding as- sumed to be present in the data. To do so, we
20
error
10
A MSE in %
trained multiple fair regression models on the
real-world dataset for increasing sensitivity pa-
rameters [I, 4]. Figure 12 we show the
relative increase in performance loss (A MSE)
10
compared to the naively fair prediciton model,
ie., r = 1. We observe only very small in- crease in performance loss, which demonstrates the robustness of our framework.
1.0
1.5
2.0
2.5
3.0
3.5
4.0
Sensitivity parameter [" (training)
Increase in prediction loss (MSE) in percent for increasing sensitivity parameter cho- sen for training the fair regressor. Mean and stan- dard deviation calculated over runs.
Figure 12:
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J EXTENSION TO CONTINUOUS FEATURES
For simplicity, we have introduced our framework framework for discrete features in the main paper. Nevertheless, our framework is directly applicable to continuous features. The derivations of the bounds are straightforward; they replace the sum over all realizations of with the integral. For the direct effect, it follows that
DE (y a.) P(as) / Σ
p ly m, m, a,) a,) (m 2()))(=))
MEM
(73)
P(aş)
/ Σ p(y|mm,,,s))PPP 2.0:)p(2)dz P as),
P(a.)
where p(2) denotes the density of Z. The other effects follow in the same manner.
To show the applicability of our framework for continuous features, we perform binary classification on dataset with multiple continuous features. The dataset is generated
Z1 Uniform(0.5 0.02Use 1.5 0.02Uspl
22 ~ Uniform[1 [ 0.02Usp, 2 0.02Use]
73 ~ Uniform[1 [1.5 0.02Use, 2.5 0.02Use]
Z4 ~ Uniform 2 0.02Usp. 0.02
(ernoulli( 0.10 +.05UUs 0.252 10.252+++//0 +,22
0.25Z3 0.524))
M ~
0.574 + 24 - 0.10RE)
= (10..22-00.22-00122000----------
for UDE, UTE, USE WV with [1, 4].
We train the three classifiers (analogous to our main paper): (1) standard, (2) fair naïve, and (3) fair robust (=our). We train them on the above datasets with sensitivity parameter & 2 and fairness constraints of 0.5 on all effects. For simplicity, we optimize the constraint classifiers with fixed penalty constraint with weight
instead of augmented lagrangian optimization.
we present bounds on the three classifiers for sensitivity parameter 2.0. We observe that training the classifier via our framework successfully restricts the bounds be lower than the constraint level of 0.5. In sum, the results confirm the effectiveness of our framework also in settings with continuous features.
In Figure 13
Munter 2n
Indtece effict
affect
Corfoanding lovel
Donfounding loval
Confounding lavel
Figure 13: Fairness bounds on the three classifiers experiments with multiple continuous con- founders.
We report the performance of the prediction models in Table We observe that fair robust model only suffers from a low loss in prediction performance while achieving faimess below the threshold of 0.5 for all three effects.
Beyond continuous features, it may also be interesting to use our framework for continuous medi- ators. Deriving sharp bounds for continuous mediators in sensitivity analysis is highly non-trivial. Therefore, no non- parametric method for such problem has been developed so far. Nevertheless, this posits an interesting direction for future research. As such, we suggest the following solution approaches:
Discretization: Discretization is commonly used in causal inference and policy learning to handle continuous variables (c.g., actions in reinforcement learning are commonly dis- crete yield continuous values). Discretization may induce overall performance loss,
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Table 6: Performance of the prediction models the continuous datasets measured by the
ROC AUC and the fairness utility.
ROC AUC
Fairness utility
Experiment
Standard
Fair naïve
Fair robust
Standard
Fair naïve
Fair robust
PUR Une UUe
1.0000
0,9921
0.7048
0.5360
0.0094
0.0302
UpEE UUR: UUge = 1.0000
0.9326
0.9774
0.5361
-..0669
0.0737
PUDE PUR PUSE = 1. 1.0000
0.9383
0.9647
0.5356
-0.0339
0.1158
PUDDE PUR PUse 1.0000
0.8783
0.9536
0.5356
-0.0723
0.1072
yet, recently, there have been several methods for handling and minimizing this loss (e.g.-
Rajbahadur et al. 2021). For certain machine learning models, discretization can even
increase the model performance (e.g.. Lustgarten et al. 2008 Varghese & Sundari 2022).
Proxy variables: Another idea is to employ proxy variables instead of sensitive attributes to enforce fairness (c.g..(uuple Gupta et al. 2018 Kilbertus et al.) 2017) or learning causal effeets
based proxies circumvent unidentifiability due unobserved confounders (e.g., Miao et al. 2018) Xu Kanagawal 2021). Hence, one solution to incorporate continuous mediators into framework through discrete proxy variable of the mediator.
37


Reviews:
[{"review_id": "E4GDjomUM0", "reviewer": "Reviewer_rYvN", "summary": "The paper integrates a recent framework for sensitivity analysis within a common causal fairness family.", "questions": "This is. summary of the above.\n\n- What is the novelty?\n\n- Why averaging over $Y$ in Eq. (11) makes sense (and which distribution are we averaging over)?\n\n- Is there any point in prescribing sensitivity analysis between $Y$ and $A$, given that $Y$ is not involved in the prediction?\n\n- Explaining better the meaning of Eq. (33) and where it is used.\n\nI would also would like to know whether any path-specific effect could be tackled and not only the three default ones.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "The paper is nicely laid out and mostly easy to follow. The problem is important, as sensitivity analysis plays a substantive role in a variety of causal estimands. End-to-end learning to produce fair classifiers is presented, illustrating that sensitivity analysis doesn't need to take place just for standard causal estimands such as average causal effect.", "weaknesses": "May main comment is to what extent this mostly is a direct application of Frauen et al. (2023). If I understood it well, is the main technical innovation the use of Lemma 1 in the context of GMSM? Or is this even more closely related to the cited paper?\n \nI’m confused by Eq. (11): there is an expectation over $Y$. Why is this form of aggregation sensible? I’d expect that we would like to enforce the constraint uniformly over all possible values in the same sample space of $Y$ instead of  performing any sort of probabilistic averaging. Speaking of which, which distribution are we marginalizing over, is this the marginal distribution of $Y$? What happens to $a_i$, $a_j$ from (1)-(3)?\n \nMaybe this is well-explored in the causal fairness literature, but the switch between “real Y” and $f_\\theta$ in the training procedure in Section 5 is very confusing: we design $f$, so any confounding between “$Y$ “ (that is, $f_\\theta$) vanishes by design. Why would we consider any sort of sensitivity parameter for “$Y$”?\n \nI’m worried about the theoretical results. In Appendix D.3 the authors state that they “extensively use the following corollary”, which boils down to Eq. (33). This equation refers to “$P(X |do(A=a), A\\neq a)$”. The event past the conditioning bar has probability zero, as the control signal $do(A = a)$ implies $A = a$ (yes, we can condition on measure-zero events, that’s what we do for continuous variables, but even then we take for granted the careful textbook measure-theoretical characterization of this conditional that we implicitly accept by default, but that’s a non-trivial (and not unique!) characterization of conditioning. But  (33) doesn’t appear to make sense even for discrete $A$). I may be missing something obvious, in which case I will appreciate a clarification – if this indeed “follows directly from basic probability theory” I’m happy to be taught the baby steps explicitly…  Of course we can say that \n\n$$P(X | do(A = a)) = P(X(a)) = P(X(a) | A = a)P(A = a) + P(X(a) | A \\neq a)P(A \\neq a) = $$\n$$P(X | A = a)P(A = a) + P(X(a) | A \\neq a)P(A \\neq a) , $$\n\nbut $P(X(a) | A \\neq a)$ is not the same thing as $P(X | do(A = a), A \\neq a)$. Among other things, $X$ is not $X(a)$ when $A \\neq a$, but it is undefined what $X$ even means under regime $do(A = a)$ and event $A \\neq a$.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "48OBFOi0kd", "reviewer": "Reviewer_2FkR", "summary": "The paper establishes the partial identification bound for causal fairness under GMSM, which can be used for learning causally fair predictors under specified GMSM and SCM. Authors provide theoretical guarantee for the partial identification bound and demonstrate its benefit using synthetic and real-world data.", "questions": "See weakness.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "I find the paper interesting and easy to read. It establishes the partial identification bound for causal fairness under GMSM, which can be used for learning causally fair predictors under specified GMSM and SCM. \n\nI believe the problem studied is important and has important practical value and the paper is theoretically sound and made a step toward better estimation of causal fairness.", "weaknesses": "The motivation and solution seem disconnected. Authors suggest practitioners can audit the unobserved confounding in the data in the introduction, but the solution using GMSM requires prior knowledge of UC strength. In this case, we need to know the unobserved confounding strength in order to estimate or optimize the causal fairness. \n\nThe novelty is relatively low. The causal fairness and the partial identification bound under MSM/GMSM is extensively studied, the paper simply combined them. \n\nAuthors did not report accuracy measures in the paper, which should also be reported. I assume such fairness constraint would impact the prediction accuracy a lot. \n\nA natural criticism is the method is restricted to discrete features, while in practice, variables like crime history may be highly contextual (videos, images, text, ..) . Can authors discuss the difficulty in extending it to high dimensional features? \n\nCan authors explain more on \"Fairness bounds in non-identifiable settings\": why general non-identifiability in these papers does not encompass the unobserved confounding case? Nabi & Shpitser, 2018 also uses unobserved confounding as one of the examples. \n\nTypo: abstract, sources of unobserved confounding? \"ours is the first work to study causal fairness under observed confounding.\" miss-specification. \n\nWriting: Why repeat the contribution at the end of section 1. The research gap seems to repeat contribution again and such gap is already mentioned before.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "zl8ncfQ6pO", "reviewer": "Reviewer_uvD6", "summary": "This paper delves into the sensitivity analysis of causal fairness criteria, specifically focusing on counterfactual direct effect (Ctf-DE), indirect effect (Ctf-IE), and spurious effects concerning unobserved confounding. The authors establish bounds for these measures by utilizing the generalized marginal sensitivity model (GMSM) and present a model for learning fair predictions. Experimental results underscore the method's effectiveness to some extent.", "questions": "The authors state that \"A key benefit of the GMSM is that it can deal with discrete mediators and both discrete and continuous outcomes.\" It is not convincing. It would be helpful to elaborate on other reasons for adopting the GMSM and discuss the strengths and weaknesses of alternative sensitivity models.", "limitations": "", "rating": 5, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "- The sensitivity analysis on unobserved confounders on causal fairness criterion is a relevant and important research problem.\n\n- The experiments on synthetic data and real data demonstrate the effectiveness of the proposed method at some extent.", "weaknesses": "- Limited Contribution and Novelty. The contribution and novelty of this paper may be limited. The theorem presented in the paper, Theorem 1, appears to be a specific application of the Generalized Marginal Sensitivity Model (GMSM) introduced by Frauen et al. (2023), which offers a comprehensive framework for causal sensitivity analysis under unobserved confounding in various settings. It would be beneficial to clarify the distinct contributions and challenges of this work compared to Frauen et al. (2023).\n\n- Limited Scope in Fairness Notions. The paper is somewhat misleading in its claim to perform sensitivity analysis on causal fairness under unobserved confounding. In fact, the focus of this paper is confined to specific causality-based fairness notions based on counterfactual direct effect (Ctf-DE), indirect effect (Ctf-IE) and spurious effects. However, causal fairness encompasses a broader range of notions, such as ones based on proxy discrimination, path-specific causal effects, path-specific counterfactual effects (including natural direct or indirect causal effect), etc. Additionally, it is misunderstanding to call the definitions Ctf-DE, Ctf-IE and Ctf-SE in Zhang & Bareinboim (2018a) as ‘path-specific causal effects’, which differs from their formal definition [1], thus leading to confusion.\n\n- Incomplete Literature Review. The paper lacks a comprehensive review of prior literature on sensitivity analysis on causal effects to unobserved confounding, such as marginal sensitivity model. Additionally, it would be beneficial to provide a brief overview of sensitivity analysis models, including the GMSM, and discuss their strengths and weaknesses.\n\n- Enhanced Experimental Analysis. In the experimental section, it would be advantageous to report and compare prediction performance across various models under different levels of confounding. This would provide a more robust assessment of the proposed method's performance.\n\n- Handling Continuous Variables. The paper assumes that the variables Z and M are discrete, yet many real-world variables are continuous. Therefore, it is essential to discuss how the proposed method can be extended to accommodate continuous variables.\n\n[1] Avin, Chen, Ilya Shpitser, and Judea Pearl. \"Identifiability of path-specific effects.\" (2005).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (poster)