Path-Specific Counterfactual Fairness via Dividend Correction
Authors: Daisuke Hatano
Satoshi Hara
Hiromi Arai
Venue: Accepted by TMLR ()

Abstract: Counterfactual fairness is a fundamental principle in machine learning that allows the analysis of the effects of sensitive attributes in each individual decision by integrating the knowledge of causal graphs. An issue in dealing with counterfactual fairness is that unfair causal effects are often context-specific, influenced by religious, cultural, and national differences, making it difficult to create a universally applicable model. This leads to the challenge of dealing with frequent adaptation to changes in fairness assessments when localizing a model. Thus, applicability across a variety of models and efficiency becomes necessary to meet this challenge. We propose the first efficient post-process approach to achieve path-specific counterfactual fairness by adjusting a model's outputs based on a given causal graph. This approach is model-agnostic, prioritizing on flexibility and generalizability to deliver robust results across various domains and model architectures. By means of the mathematical tools in cooperative game, the Möbius inversion formula and dividends, we demonstrate that our post-process approach can be executed efficiently. We empirically show that proposed algorithm outperforms existing in-process approaches for path-specific counterfactual fairness and a post-process approach for counterfactual fairness.

Published in Transactions on Machine Learning Research (03/2025)
Path-Specific Counterfactual Fairness via Dividend Correc- tion
Daisuke Hatano
dattoku.hatano@@gmad.
RIKEN Center for Advanced Intelligence Project
Satoshi Hara
satoharaShuec.acc.p
The University of Electro-Communicatii
Hiromi Arai
hiromi.arai@riienn,
RIKEN Center for Advanced Intelligence Project
Reviewed on OpenReview: https: // optp://eeee..eee...mm net/ formm? id= RXoSmsyObR
Abstract
Counterfactual fairness is a fundamental principle machine learning that allows the anal- ysis of the effects of sensitive attributes in each individual decision by integrating the knowl- edge of causal graphs. An issue in dealing with counterfactual fairness that unfair causal effects often context-specific, inftuenced by religious, cultural, and national differences, making it difficult to create a universally applicable model. This leads to the challenge of dealing with frequent adaptation to changes in fairness assessments when localizing model. Thus, applicability across a variety of models and efficiency bocomes necessary to meet this challenge. We propose the first efficient post-process approach to achieve path- specific counterfactual fairness by adjusting a model's outputs based on given cansal graph. This approach model-agnostic, prioritizing on flexibility and generalizability to deliver ro- bust results across various domains and model architectures. By means of the mathematical tools from cooperative game theory, the Möbius inversion formula and dividends, we demon- strate that our post-process approach can be executed efficiently. We empirically show that our proposed algorithm out perporroms existing in-process approaches for path-specific coun- approach for counterfactual fairness.
terfactual fairness and a
1 Introduction
The rise of machine learning has significantly transformed decision making processes in daily life. However, in high-stakes applications such healthcare (Ahsan et al. 2022). criminal justice (Brennan al. 2009 hiring (Hoffman et al.) 2018), and lending [Khandani et al. 2010), fairness becomes a critical consideration.
As these domains directly impact individuals, it is essential to ensure that decisions made by machine learning models are both fair and unbiased. Recently, the machine learning community has developed various statistical criteria to formalize fairness notions (Dwork et al., 2012: Feldman al. 2015 Hardt et al. 2016 However, most criteria. are enforced by introducing fairness constraints during the model training process, often creating a trade-off between fairness and accuracy.
To relax this trade-off, Kusner et al. (017)) introduced counterfactual fairness, which aims to ensure fair decisions without sacrificing accuracy too much by focusing on the causal relationship between a sensitive attribute and an outcome. Their framework assumes that any causal effect of the sensitive attribute on the outcome as inherently problematic and aims to eliminate any influence that the sensitive attribute might have on the outcome. However, failing to distinguish between fair and unfair pathways can lead the removal of legitimate effects. This oversimpiifiatton risks undermining the model's fairness. In cases involving physical strength, outcomes such as suitability for physically demanding jobs may depend

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sensitive attributes due to biological differences. Removing the causal path from gender to physical strength
could result in unfairness. This complexity highlights the limitation of conventional fairness notions, which
struggle to distinguish between fair and unfair causal pathways. The fairness in such scenarios cannot be
adequately captured using traditional statistical metrics like demographic parity in group fairness (worr
iet al., 2012; Verma& & Rubin 2018) or even counterfactual fairness. To address this issue, Chiappa (2019)
Wu et al. (2019b) introduced path-specific counterfactual fairness, which aims to climinate only the unfair
causal effects of the sensitive attribute while preserving the fair ones. This framework has inspired range
of fairness algorithms designed to achieve path-specific counterfactual fairness (Zhang et al. 2017 Nabi &
Shpitser 2018
Chiappa 2019
Wu et al. 2019b; Chikahara et al. 2022).
In this study, we propose a posl-process method to achieve path-specific counterfactual fairness. A significant challenge in achieving path-specific counterfactual fairness is that unfair pathways can be influenced by specific religions, cultures, or countries, making it difficult to develop a universally applicable model. For instance, fairness asscssments often depend on regional ethical standards, such as General Data Protection Regulation (GDPR) D in Europe, with different. standards prevailing in other regions. This variability requires adaptations of fairness assessments to local contexts, highlighting the need for an efficient and flexible fairness method. In contrast, most counterfactual fairness methods roly on in-process approaches, which require building bainess-aware models from scratch. To overcome this limitation, we propose a method that achieves counterfactual fairness by post-processing the outputs of existing models, enhancing the adaptability of fairness assessments across diverse cultural and religious settings.
In our method, we leverage the set function framework, as the predictive model's output can be interpreted as set function when the sensitive attribute is binary. As conventional set functions cannot accurately capture the causal effect of the sensitive attribute on the output, it is essential to implement framework that can isolate causal effects from correlations. By introducing an interior operator, we construct a causally-aware set function that effectively captures causal dependencies. We then clarify the relationship between the causally-aware set function and path-spocific counterfactual fairness. To deconstruct the causally-aware set function into meaningful parts, we introduce a specialized Möbius inversion formala (Stanley) 2011 Algaba et al. 2004) that accounts for hierarchical constraints that reflect causal relationships. This enables us decompose it into the sum of dividends (Harsanyi 1958). The dividend can be intuitively understood additional value created by modifying specific attributes, after considering the overlapping contributions of their smaller attribute subsets. We elucidate each dividend corresponds to well-known causal effect and achieve fairness by removing dividends associated with unfair pathways.
One of our contributions is to propose the first post-process approach designed for path-specific counter- factual fairness. To this end, we introduce a novel causally aware set function by leveraging concepts from combinatorics to handle precedence structures. Through the decomposition of this set function, we eluci- date the relationship between its components and various causal effects, corresponding to total direct effect, path-specific effect, and mediated interaction. This decomposition enables more precise analysis of how individual causal pathways contribute to the overall effects. Another contribution is that the execution time of our algorithm is linear with respect to the size of the given causal graph, as it is based on running the model inference a number of times proportional to the number of nodes in the graph. Additionally the al- gorithm requires model training time for each mediator. Despite this additional computation, our algorithm remains efficient compared to the existing in-process approaches, as shown in the experimental evaluation.
2 Related work
Approaches for ensuring fairness in machine learning models typically fall into three main categories. (i) Pre-processing approach entails modifying the training data to remove biases before the model training stage, e.g., by group-wise data rescaling (Feldman et al. 2015), data re-labeling {Kamiran & Calders 2012), and instance re- re-weighting [Kmmiran & Calders 2012). This approach is model-agnostic because it focuses on the input data rather than the structure of the model. However, pre-processing alone may not fully satisfy the fairness condition in the final model prediction. (ii) In-processing approach adapts the learning algorithm to prevent bias, e.g., by incorporating regularization terms to penalize unfair outcomes (Kamishima et al.,
https://eur- www.eeee.errreeeeee eurgpa..e/eel/rre/20/////////
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Fukuchi & Sakuma 2014 or by introducing fairness constraints (Zafar et al. 2017 Donini et al, 2018 A Agarwal et al. 2018). In-process approaches tend to be more complex and computationally expensive, but can directly control biases in the learning process. They are often limited to models that support gradient-basee optimization, making them difficult to apply to certain models, such as dccision trees. (ii) Post-processing approach adjusts the predictions of the trained model to ensure fairness by modifying the outputs (Kamiran et al., 2012: Hardt et al. 2016). This approach is computationally efficient and flexible, as it modifies the output of given model and can be applied any learning model after training. However, post-processing approaches often compromise accuracy more than in-process approaches.
2012
Fairness studies within the context of causality have mainly concentrated on in-process approaches. Most fairness criteria related to causality, such as counterfactual and path-specific counterfactual fairness, have been developed within this framework. Some approaches try to ensure the fairness by integrating fairness criteria into the model's optimization process by adding fairness constraints during training (Kusner et al., 2017 Nabi & Shpitser] 2018 Wu al., 2019b; Chikahara et al., 2021). Another approach focuses on modilying individual data points to ensure fair outcomes (Chiappa 2019 Applying in-process approaches in settings where unfair pathways require frequent modification presents several limitations: (i) Adding fairness constraints during training makes the model optimization process more complex, which increases the time required to train models. (ii) Fairness constraints restrict a model's flexibility, especially when certain learning models cannot easily accommodate these constraints within their optimization framework, such as decision trees, k-NN, clustering algorithms, and ensemble models that lack gradient- based optimization frameworks. Post-prost Post-process approaches can overcome these limitations and do not alter the model training phase, allowing for flexibility and computational efficieney. The only post-process approach was proposed by Wu ct al. (2019a) to address counterfactual effects, but it is limited to counterfactual fairnoss, and the computational complexity of their LP-based approach increases exponentially with the size of a causal graph. In contrast, our research presents novel approach for achieving path-specific counterfactual fairness that runs in linear time relative to the size of the causal graph.
Our main mat hematiiaaall tools, the Möbius inversion formula and dividend, are widely used in the field of co- operative game
AAgaba et al. 2004 Ieong & Shoham 2005). These notions are closely related the Shapley 1953) which is now widely used in machine learning, c.g., for feature selection (Cohen et al.
value (Shapley
2005
Sun et al.) 2012), data valuation (Jia et al. 2019 Ghorbani & Zou| 2019). and explainability (Lundberg
& Lee 2017
Sundararajan & Najmil 2020) (see [Rozebberczki et al. (022)). Although these mathematical
tools are from the same field, the way we use the Möbius inversion formula and dividend are different from the aforementioned studics. While we use the Möbius inversion formula and dividend for post-hoc model correction, Shapley value is used to quantify the contribution of each individual (such as feature or data instance) in the aforementioned studies. Our finding will provide new possible connection between machine learning and cooperative game theory beyond the Shapley value.
Feasible sets are a notion introduced in Antimatroids (ortee et al. 2012). and it also has been studied in conjunction with the Möbius inversion formula (van den Brink 2017 Algaba et al. [004). Antimatroid is used as a hierarchical structure among players called permission structure (van den Brink & Gilles) 1996; van den Brink 1997). and the Möbius inversion formula is used as a tool to analyze the properties of the Shapley value. To our knowledge, we will be the first to employ antimatroids and the Möbius inversion formula in the analysis of causality.
3 Preliminary
3.1 Path-specific counterfactual fairness
We introduce key notations to formalize concepts in causality and path-specific couttrrfactual fairness. Causality is often expressed using a directed acyelic graph, known as a causal graph. Let G be a directed acyclic graph, where a set of nodes corresponds to features X, and a set of dirccted edges indicates cansal relationships between them. For each direct edge i } where i,j X, i is referred to as a parent of j while is referred to as child of i. The set of nodes that can reach node i is referred to the ancestors of i.
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Figure disppaass example of causal graph illustrating hiring decisions for physically demanding positions. In this graph, the endogenous variables Q, D, M E X represent gender, qualification, faculty type, and physical strength, respectively. The qualification represents the skills and knowledge, the faculty type refers to the specific academic department, such as STEM (Science, Technology, Enginecring, and Mathematics) fields, and the physical strength describes an individual's physical ability or capacity. Gender is a sensilive aftribute, while the features and M are referred to as mediators, which are influenced by A through cansal pathways. Nodes ep and eM indicate exogenous variables that work as error variables. For simplicity, we do not show exogenous variables for each mediator in the following causal graphs. In the path-specific counterfactual setting, not all influences from the sensitive attribute are necessarily unfair. In Figure 1 the causal path through A + M + Y is considered fair, as it reflects a relationship influenced by innate factors between the gender A and the physical strength M. Conversely, the pathways A + Y and A + D + Y are considered unfair, as they involve social or cultural factors that may lead to discriminatory outcomes. For instance, traditionally, men tend to enroll in engincering, physics, and computer science programs, while women are more inclined to choose fields such as nursing, education, and social work. Note that every edge in the causal graph belongs to either fair or unfair pathways. While the fairness of pathways may frequently change depending not only on the location where the predictive model is applied, but also on the specific context or situation, the unfair pathways in this setting are fixed.
We concentrate on path-specific counterfactual fairness as out- lined by (Chiappa 2019 Wu et al.) 2019a). which seeks to iso- late the effect of unfair pathways. The set of unfair pathways is
qualification
strength
denoted as n; L.e., x = {A + Y, A + D - =-Y, V} in the example. To establish the definition of path-specific counterfactual fair- ness, we will exainne path-specifie causal effects as described by {Avin et al.) 2005 Albert & Nelson 2011) that measures the extent to which an attribute influences other features through causal paths. The path-specific causal effect can be obtained by comparing two predictions generated by modifying inputs X. The focus is on situation in which an individual accepts the aforementioned fairness while being assessed for hiring based on the prediction of a machine learning model. Let Sn and JM be causal models of the mediators D and M from which the values of D and M can be estimated, i.e., D = fD(A, (A,A,,,ep) and M = (A, ear) In addition, the value of Y can be obtained using the predictive model f. given by Y = f(A, Q. D. M). Assuming that the value of A for a given data point X is O, the path-specific counterfactual effect (PSE) can be defined as the difference between two predictions YA=1]|= YA=o, when the value of the sensitive attribute A flips from 0 to 1 under unfair pathways n, where Yf and YAVO are called potential outcomes, expressed follows:
gender faculty
decision
A
M
Y
eD
eM
Figure 1: An example of a causal graph in- dicating hiring decisions for physically de- manding jobs (Chikahara et al. 2022). The two red paths denote unfair pathways.
YA=1I = 5 (1,21 (00 eM),, Yse = S(0, Q. (en)) Q, eM)))
where Yl= represents a prediction when modification of A only affects unfair pathways, implying that the value of M remain unaffected by the change. Specifically, this means M f0, ex))) On the contrary, as Y represents the prediction for the original data point X, it is not influenced by æ. Employing these notations, path-specific counterfactual fairness can be defined using PSE, given as follows:
Definition 3.1 (Path-specific counterfactual fairness (PSCF) CChiappa, 2019 Wu et al. 2019b) ). Given
the unfair pathways π of a causal graph G and causal models fo and fM, a classifier f achieves path-specific counterfactual fairness if for any value x of X, it holds that
YY (V - YA 1 X = X= x)==0 =0
(1)
Since the PSCF designed for binary sensitive attributes, needs to be generalized to accommodate non- binary sensitive attributes. For simplicity, we assume in the following section that the value of the sensitive attribute is flipped from 0 to 1 without loss of generality.
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3.2 A decomposition of the total effect into path-specific effects
Using the notations of Figure 1 we introduce some causal effects and some of its decomposition. The total effect of alternation on the sensitive attribute A is defined as
TE = -E==Ya--YYa-==
f(0,, (.0
For simplicity, we will write [(A, Q. (A ep)): Q. ear)) as YADAMA. since Q, CD, and eM are not affected by the change in A. TE can be decomposed as TE {YiDi M2 YOD,M, (YoD,M, YoDo where the first term is the total direct effect, denoted by TDE, and the second term is the pure indirecl effect, denoted by PIE (Robins & Greenland 1992 Vander
2013 TDE is the effect of flipping sensitive attribute A from O to 1 while considering the effect of changes in A on the mediators. This allows us to capture the interaction elfect between A and the mediators D and M. In contrast, PIE is the effect of changing the values of the mediators while keeping A to This means PIE represents the indirect effect, excluding the interaction effect between A and the mediators and M.
PIE can be further decomposed into path-specific effects. However, Avin al. (2005) showed that the sum of path-specific effects is not equivalent to the total effect, certain components are missing. Taguri et al. (2018) later proved the missing components are due mediated interaction. By introducing this effect, PIE can be decomposed into the sum of path-specific effects and mediated interaction, given by:
PIE == (YOD, - YOD,M, } + (YoD, - YOD, Mo - (YoD, M. - YOD, - YODOM, + YoDaMob,
(2)
where the fist. term the path-specific effect via the pathway A + D Y, the second term is the path-specific effect via the pathway A - M + Y, and the third term is the mediated interaction between A D and A + M. The path-specific effect measures the effect of an intervention an outcome along specific causal path involving mediators. For pathway of A + D - Y and target mediator D, the path-specific effect can be calculated as the difference between the predicted values when M takes on a specific value influenced by the the exposure and the predicted value when M is set to its original data point. Mathematically, the path-specific effects via mediator D and M are defined by
PSEp == YOD, M, - YoDam, PSEM := YOD,,, YOD, Mo:
respectively f
The last term of equation 2 is referred to the mediated interaction, given by
MIрM == -YYD,, YOD, YOD,M, + YoDoMo).
Then PIE can be represented as PIE = PSEp + PSEM + MMDM, and we have
TE = TDE + PSEp + PSE + MI
4 Problem setting and motivating scenario
The motivating scenario is the release of an Al-based focision-making system that involves two main stake- holders: an AI model publisher and an auditor. The publisher aims to deploy an Al-decision making service built on their predictive model f. The auditor's role to assess this model to determine whether it meets fairness critcria. To ensure the integrity and transparency of the service, the publisher must publicly dis- close the causal graph G that underpins the predictive model f. The auditor will evaluate the fairness of f based on this pre-registered causal graph G, regardless of its accuracy. In addition, the auditor should learn the causal models of mediators, such as fo and fM, to accurately estimate their values when altering the sensitive attribute. This is important because, when the publisher can define their causal models, the values of mediators can be adjusted as desired when altering the sensitive attribute.
For instance, if the publisher sets up a model where the sensitive attribute (gender) influences the mediator (income), they can adjust how much income changes when the sensitive attribute flips from male to female,
2PSE is alternatively defined PSEp = VID,Mo VIDwMo or VID,M, VIDDM, in Taguri al. 2018 Huang & Cal 2016 Huang& & Yang 2017 Steen 2017
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This adjustment can lead to manipulated fairness outcome, the publisher has the ability to tweak the system to make it appear fairer or more biased based on how they model the interaction between gender and income. This ability to influence the intermediate mediators could distort the auditor's ability to evaluate fairness accurately.
Therefore, this paper, we define the following problem setup: (i) A causal graph G, unfair pathways n, and predictive model f are given, and (ii) Causal models of mediators, e.g., fo and fM, need be prepared by the auditor. With these settings, our objective is follows.
Objective. Our aim is to develop a post-process approach that modifies the output f(x) of a given model f for every individual data 2, such that equation holds.
5 Proposed algorithm
We first briefly describe the idea of our algorithm.
The key components of our algorithm is feasible sets, dividends, and Möbius inversion formula. The feasible sets represent subsets that capture causal relationships, and the dividends quantify additional contribution by modifying specific attributes, accounting for the overlapping contributions of their smaller attribute subsets. The Möbius inversion formula serves as the method for computing these dividends. The proposed algorithm operates through five steps, as shown in Figure 2 (i) A prediction f(x) of data point x is translated into set function UF that incorporates causal relationship, as focus the binary state of sensitive attribute. To achieve this, we employ feasible sets to represent causal relationships and an interior operator to incorporate the feasible sets into the set function. (ii) The value of uy is decomposed into the causal effects of individual causal paths through the Möbius inversion formula, allowing us to decompose the value into the sum of dividends. We elucidate that the dividends of any subset in UF correspond to TDE, PIE, or MI, as described in Section 3.2 (iii) Using this relationship, fair dividends A.G are obtained by nullifying dividends Any of every unfair pathway. (iv) A fair set function v'z- can be constructed from the fair dividends Aoy, exploiting the property of the Möbius inversion formula, which ensures any set function has unique representation in terms of dividends. (v) Ensure that the value d's obtained in Step (iv) corresponds to the desired value f((x), given that the translation from set function to prediction on x can be performed without loss.
5.1 Construction of causally aware set functions
The aim of this section is to create causally aware set function ey, using a given predictive model f and particular data point x, which is related to Steps (i) and (v) in Figure ? We construct uy that satisfies the following condi- tions: temporal precedence and unconfounded- ness. These conditions are necessary for inferring causal effect, as described in Pearl (2009). Tem- poral precedence ensures that the cause occurs be- fore the effect, which required for establishing a causal relationship. Unconfoundddness ensures that there are no unmeasured confounders that influence both the cause and the effect. In other words, it means that any confounders must either be measured accounted for. Unfortunately, conventional set functions do not hold the above properties. In the following, we describe how to construct causally aware set function uy to satisfy the desired properties.
prediction
set function
dividends
f(x) Step (i)
U/F
Step (ii)
UF
Step (fii)
Step (v)
Step (iv)
('(x)
fair prediction
fair set function
fair dividends
Figure The flow of our algorithm.
We begin by creating a conventional set function Us.r 2 + R of f(x), where N stands for a set of n features in X excluding the target variable and æ and r indicate feature values of a given data point and a point of reference, respectively. For a given subset S C N, the value (S) can be represented by analyzing the impact of the absence of features in the complement set N N given by (S) == f frs. for any S C N, where is n-dimensional vector called reference value, and #ss) indicates n-dimensional vector
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whose i-th component is ry if E S and z₁ otherwisd" S represents the complement of S, i.., S = NN S. The value varr (N)) indicates the prediction obtained by changing the value of i from to ri. In what follows, unless otherwise specified, we will abbreviate Ugrr as v.
The set function v constructed above is not compatible in causal contexts because it does not accurately present causal relationships. To implement the relationship into the set function, we employ an interior operator (Korte et al., 2012), given by int; : 2 + F, where F is a family of subsets of N referred to feasible sets, which is often referred to as ancestral sets in the causal context. Every feasible set S E F satisfies the requirement that if a node is included in S, then all its parent nodes are also into S. The interior operator is formally defined by
int int := UITETETSTTTE V
(3)
With these notations, causally aware set function by is defined as follows:
(y(() := (S)) C...
(4)
Intuitively, the interior operator modifies an infeasible condition, where some children absent features are present, into a feasible condition, where no children of absent fcatures are present. Hence, the causally aware set function oF has a capacity to identify causal relationships.
Example 1. Feusihle sels are given by F {0, (A): (Q), (A.,), D), (A, (AQ. M]} in Figure 4 In Figure 3 (b), given {Q, M3, the interior operator returns {Q}, and the value of uy (1Q, DyM corresponds to v((21) f FD.
Figure 3 (a) demonstrates the operation of a set function v and an interior operator when N \ {4} (Q, D. M) is given. Evaluating the value of (Q, D, M) on does not reflect. causality nodes D and M are children of the absent feature A. This implies that the values of the nodes D and M remain unchanged unless the parent's value A changes. Alternatively, the interior operator provides feasible set {Q), which exhibits a causal relationship.
The reference value r plays an important role in the causally aware set function VF to measure the causal effect size. The reference value accounts for how changes in the sensitive attribute influence the values of other features, ensuring a proper estimation of causal effects. Since the true causal model is not known, we need to learn the mediators' models fD. far to accurately estimate the reference values. The causal models fn and JM are learned through the following process. First, we estimate the value of the exogenous variables for each data point Then we use these estimates to learn causal models for the mediators, For estimating the exogenous variables, we can apply several estimation algorithms, such as Maximum Likelihood Estimation (MLE) via the Expectation innetaationMaxiiizaatioon (EM) algorithm and Bayesian inference using methods like Gaussian approximation or the Markov chain Monte-Carlo (MCMC) method. The detailed approach used in our experiments is written in Section C.3 After estimating the value of exogenous variables, causal models for mediators can be learned using appropriate lcarning models, such linear regression neural networks. As a result, the reference values for the mediators can be obtained from the learned causal models and the estimated exogenous variables. For variables that are not affected by the sensitive attribute, their reference values remain unchanged from the given data point x.
In Figure 1 by changing A from 0 to 1, the values of D and M are also affected. They determine their reference value by eramining their causal models fo and fxs, which are obtained by T = 1, TD = =)), and FM = (r (TQ)- On the contrary, as the value of Q is not affected by the value of A, and its reference value remains unchanged, i.e., TQ = JQ.
Example 2.
The causally aware set function ez satisfies the two conditions of temporal precedence and unconfoundedness. Temporal precedence implies that a cause occurs before its effect. We can easily check that Uf supports the temporal precedence from Figure 3 since the existence of child nodes, c.g., D and M, depends on the existence of their parent nodes A and Q. In other words, changes in A or Q lead to changes in their children,
3 Although the set. function is defined as (S) == ((55)55)) [Lundberg Lee 2017 in this paper, we swap the positions and r for convenience.

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TQ
rQ
AA
A
TD
(a) o(N {A)) fI TO,
(b) (F - =(aa))
Figure 3: How the conventional set function and the causally aware set function by work.
D and M, indicating causal direction. Unconfoundenness means that the non-existence of unmeasured confounders. In our problem setting, the reference values of any confounder are equivalent to its data point æ. This implies that for any S N and any confounder E N \ S, Uy(S) = (x((U This aligns with ignorability by satisfying that confounders outside of have no impact on the causal effect.
In this section we have focused on changing the gender altribute from 0 to 1. However, this approach is not limited to binary attributes; it also applies to sensitive attributes that can take more than two values from discrete set. Nonetheless, when the sensitive attribute has more than two values, it important to define the fairness criteria carefully, as they may not be uniquely determined in such cases.
5.2 Decomposition into the sum of causal effects
The purpose of this section is to clarify the connection between by and causal effects introduced in Section 3.2 The connection allows us to ensure the path-specific counterfactual fairness by removing the corresponding causal effects of unfair pathways, which falls under Steps (ii), (iii), and (iv) in Figure 2 We first show that Up can be presented through the sum of causal elfects by using the Möbius inversion formaua (Stanley 2011), to decompose an obtained set function v into the sum of dividends, which are a particular form of synergy effect.. The Möbius inversion formula of a function structured by feasible sets ey was proposod by Algaba al. (2004), given by
(S = - Aors (T), 3 N,
where Aur 2 R is called Möbius transformer of by or is often referred to (Harsanyy) dividend in the cooperative game context Haasanyi
1958).
For each T C N, the dividend Auy (7) is computed by
Σ -maat i
A (T)
(6)
0,
otherwise,
where ł : [7], : [U], and en()) is set of nodes that does not have any child in T. Intuitively, A (T) indicates the additional accuracy obtained by modifying attributes T, after accounting for the overlapping accuracy contributed by the subsets of T.
We then explore the relationship between path-specific counterfactual fairness and dividends. The lemma suggests that the PSE can be computed leveraging the dividends representation.
Lemma 5.1. Let Fx be a feasible set of DAG induced by unfair pathways a. Then the prediction v(N) is palh-specific counterfactual foir if and only if ils corresponding cousally-aware set function wy, holds that
ESEAr Aur, (S)=0) O.
Proof sketch. To connect the dividend representation and the definition of PSCF, given in equation D we define r' as the reference value affected only through unfair pathways and Fr as feasible sets of DAG induced by unfair pathways, as exemplified in Example 6 Let Uz denote a set function obtained by replacing r into r' from Vr, and Ur,r,, denote a set function obtained by replacing F into F= from ey. We show that EsEr A = Σser A Se...r.(. (S) using the binomial theorem. The statement obtained by using this relationship.
Lemma 5.1 indicates that we can obtain fair prediction by computing the dividends of all feasible sets on unfair pathways Fx and setting the value of the dividends to zero. Thus, we obtain the next lemma.

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Lemma 5.2. Given unfair pathways n, modified prediction fx) achieves path-specific counterfactual fairness if it holds that
+ (((())+EsEEAA00 Aus. (S).
(8)
Proof. We first set the sum of dividends of f'(x) as that of f(x), that is, EsEF (S). By setting the values of the dividends involving unfair pathways Fr to zero, satisfy the condition of Lemma 5.1 i.c. the value of (x) matches (a(x) Eser. Auz, (S). Hence, we achieve the desired result.
We can interpret each dividend as the notation of causal effects introduced in Section 3.2 The next propo- sition outlines this interpretation.
Proposition 5.3. The dividend - (S) shows the total direct effect if S is a root node of G, the path- specific effect if indicates single pathwaay, and the mediated interaction S is the union of multiple pathways.
Proof sketch. We can easily prove the proposition by correlating each equation of causal effects and the cal- culations of corresponding dividends. Here, we demonstrate the causal effects some specific subsots, using Example 1 The total direct effects altering the value of is given by - ((A)) = uz =0) ((A{) = YID, M, Yo TDE. When feasible set is singleton, i.e., S E F with |S| = 1, the correspond- ing dividend corresponds to TDE. The path-spocific effect through D is obtained by
(fA, D))
(f(A) 1y ((A D)) Yod, M. YUD,M, PSEp. When feasible set is constructed by a single pathway, e.g., 5 [4, (A, D). ( (A, M), the corresponding dividend is equivalent to PSE. The mediated interaction between M and D is calculated by -A, (IA, D, M)) - -(v>({A))) - (=((A,))
((A,,, D, (A,, = -(YOD, M, - MOD,N, - YOD,M, + YoD,M,) = MIDM- When a feasible set is constructed by the union of multiple pathways, e.g., S = D, the corresponding dividend is equivalent to MI. The details are provided the appendix.
e=({A,
With Lemma 5.1 and Proposition 5.3 a modified prediction f'(x) is obtained by computing the dividends of all feasible sets on unfair pathways Fx. Intuitively, these statements indicate that the prediction f(x) is path-specific counterfactual fair if causal effects from the sensitive attribute are ineffective to the prediction. Furthermore, we note that our approach can independently quantify the influence of unfair pathways by introducing specific parameters to represent their respective degrees of impact. By doing so, we enable to adjust for the varying impact of unfair pathways, ensuring & more precise handling of fairness issues and achieving a robust framework for mitigating unfairness.
5.3 Simplification of computing dividends
Unfortunately, as the size of the causal graph increases, removing the dividends of all unfair pathways and their unions using Lemma 5.1 can lead to an exponential increase in time complexity. However, the computational time can be siguificantly reduced. As shown in equation 7 it suffices to compute the sum of dividends over subsets S Fx, corresponding to unfair pathways. We can rewrite it in fairly simple form, shown in the next theorem.
Theorem 5.4. Let P Punfotr be union of all unfair pathwaays, i.e., Panfair == USEF S. Then it holds that
f((x) = f(x) + u(Panfair) v(0).
(9)
Proof. It is evident that ESEr, Aor, - uy. (Punfuir). Then we have ESER Auy, (S)= =0)=,, (Panfair) UF. (0). Since Ur, (S) returns v(S) when S E Fa and it is obvious that P Punfair E Fr, we have s'(x)
f(x) + u(Punfatr) (B), and we complete the proof.
The key insight of Theorem 5.4 is that it not necessary to treat the complex notions of feasible sets F and the dividends Aey in the Möbius inversion formula. To procoed with our algorithm, it is sufficient construct Puntair by the depth-first. search and predict v00) and v u(Punfair)
Example 3. Considering the example in Figure 1 we have Fz {0, (A), DJ), Panfair D).
Consequently, f'(x) (xx) + ((AA,)) v(0).
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Algorithm Dividend Correction
1: Input: causal graph G, causal model, a predictive model f, a data point 2, and unfair pathways
a fairness parameter E [0.1]
2: Output: modified prediction f'(x:
3: Estimate the values exogenous variables and train predictive models for each mediator.
Compute a reference value from the trained causal models.
Compute Punfair: which is union of all unfair pathways.
(( < f(z) +(v(( ((v(Punfair) =0))).
return (x)
5.4 Overall algorithm
The overall algorithm, called dividend correction, is given by Algorithm I which modifies the prediction such that the condition equation is satisfied. A brief outline of the proposed algorithm is as follows: In line B it estimates the values of exogenous variables of each mediator and predicts causal models for them using the estimated values. In line 4 it creates reference value r on the basis of the causal model to build a set function v. The reference value of a sensitive attribute is considered 1, as the original value is 0. Features that are not affected by the sensitive attribute, e.g., Q, maintain their original values, whereas those affected by the sensitive attribute, e.g., M and D, obtained their values from the trained causal models. In line 5 it computes
Punfair: used in Thcorem 5.4 from unfair pathways. The Punfair indicates the set of all nodes on unfair pathways and can be performed by proceeding a depth-first search in a causal graph G. In line G it eliminates the predictions of subsets computed in line 3 from f(x) to produce fair prediction (((x). A fairness parameter E [0, I] is included to adjust the degree of fairness, where 1 indicates that it completely removes the effects of unfair pathways.
The computational complexity of our algorithm involves the time of the union procedure to compute Punfair: the time to compute a reference value r, the time to estimate exogenous variables, and the time to learn causal models. The union procedure takes linear time based on the size of a DAG, as these are obtained by scanning a segment of the DAG consists of the unfair pathways. Tet d be the dimension of feature space and g(d) be the time complexity of inference time by causal model. The time complexity of computing the reference value takes O(N gdd)). Let h(d) be the time complexity of computing learning model. The time to compute causal models for mediators is O(N h(d)). Let be the iteration number of EM algorithm, and let m be the number of data points. e(m) N i) denotes the time complexity of running EM algorithm.
Theorem 5.5. The computational complexity of Algorithm is O(LE) + |N] (g(d) + h(d)) + e(m|N[()), where N and E are the set of nodes and edges in a DAG, respectively.
6 Experiments
We demonstrate the effectiveness of the proposed algorithm by measuring the PSE, accuracy, and runtime using synthetic, Adult (Bache Lichman 2013), and German datasets. The experiments were performed using a Macbook Pro with Apple M1 Max and 64GB RAM.
6.1 Comparison with in- in-process approaches
We first evaluate the accuracy and PSE of our algorithm compared with some in-process approaches because there is no post-process approach for path-specific counterfactual fairness settings. Recall that instances for path-specific counterfactual fairness are paramcterized by causal graph G, set of features X, and model f to predict Y. The detailed setups for each dataset are described in the appendix.
2021 Our synthetic data setup was based on the configuration used by (Chikahara et al. hiring decision scenario, as shown in Figure [ indicating a causal graph G with X
which focused on (A, D, Q,Q, Q) and
A symbolizing sensitive attribute. We used unfair pathways (A Y. A + Y) for the synthetic dataset.
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Proposed
PIU
FIO
Remove
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10a
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synth adult german
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Figure Test Accuracy vs. PSE for Synthetic (1st column), Adult (2nd column), and German datasets (3rd column), with runtime (in seconds) shown in the 4th column.
Our Adult setup was based the configuration es- tablished by (Nabi & Shpitser, 2018 Chikahara al.) 2021), for which we used the source code provided on their Github repositorie 1F As shown in Figure 4 5 attributes were examined: gender A, marital-status M, oducation L, occupation R, nationality C, from at- tributes, because gender was identified as the sensitive attribute be trcated. A direct path from gender income and indirect paths from gender to income via marital-status to unfair pathways
nationality
education occupation
marital-status gender
income
Figure 4: The causal graph of Adult
dataset (Chikahara et al. 2021 The red pathways denote unfair pathways.
established, i.e.,
( M + Y, A M + L Y,A R R- + M + + --
Our German setting was based on CChikahara et al. 2021 financial information S (including saving accounts, checking account, and housing), information about debts R (including credit amount and duration), and others C' (including age, job and purpose), as sex was defined the sensitive attribute, shown in Figure X of the appendix. We prepared three unfair paths from to each feature of financial information.
Nine attributes were examined: sex A,
Given the absence of post-processing approach specifically designed for path-specific counterfactual fair- ness, we carried out a comparative analysis the proposed algorithms against three established in-process approaches: the probability of individual unfairness (PIU) /Chikahara et al. 2021 outcome (FIO) (Nabi & Shpitser 20188, and Remove algorithmn. PIU aims to minitiiz the probability of in- dividual unfairness, and FIO tries reduce the mean unfair effect, defined in Appendix. To create the basic learning models for FIO and PIU, employed logistic regression and neural network. In addition, pared Remove algorithm that learns model without considering features that appear unfair pathways. As learning models f, we implemented logistic regression (LR), random forest. (RF), and neural network (NN) for our algorithm. Algorithms without fairness assessments can be directly implemented within our algorithm by setting = It is worth mentioning that thanks to its model-agnostic nature, our algorithm can adopt models where gradient-based learning does not work, such as R.F.
2022 fair inference of
pre-
Since the true causal models of mediators are unknown except for the syuthetic dataset, we approximated them using linear regression. We estimated the values of exogenous variables that not present in the original dataset. To this end, utilized expectation- mixittiaommaiiiiiaiioin (EM) algorithm (Dempster et al. 1977 to perform maximum likelihood ostimation in latent variable models and to estimate the posterior distributions of the latent exogenous (noise) variables, as they are not directly observable. While the noise is assumed to

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be additive, its specific values must still be estimated from the data 4 With these estimated values, we then learned causal models for the mediators through linear regression. We applied these learning models as the true causal models when computing YA=1||- in evaluating PSE. The effect of the model mismatch between the true causal model and learned causal model will be mcasured in Section 6.3
Figure 5 shows the test accuracy and runtime of algorithms using neural networks as the base model, plotted against PSE for synthetic, Adult, and German datasets. The error regions represent one standard deviation. The results for other base models are given in Figure[[]j in the Appendix. We varied the fairness- related parameters for each algorithm, except for the Remove algorithm. Each plot illustrates the average performance across 10 instances. The runtimes are averaged over the combinations of 10 instances and fairness parameters.
For Algorithm [I] we proceeded line 0 where e alters from 0 to 1 in increments of 0.05. For both FIO and PIU, their penalty parameters were modified with respect to fairness from to in steps of 0.1. Figure f shows that superior results are characterized by higher accuracy and lower PSE. A notable result here is that our post-process approach is better than the existing in in-process approaches when using logistic regression and remains competitive when using a neural network, despite in-process approaches having access to the causal graph during model training to achieve higher accuracy. Moreover, our algorithm achieves comparable accuracy while requiring significantly less runtime. Our algorithm could attain the better PSEs and runt by utilizing the causal graph only in the post-process step but not during the model learning. The result with the highest PSE (the rightmost point) indicates the accuracy without any modification. By examining the gap between the accuracy at the highest PSE and the lowest PSE, one can assess the extent to which learning model relies unfair pathways. In the case of the Adult dataset, the random forest does not rely on unfair pathways to achieve higher accuracy compared to logistic regression and neural nerworks.
6.2 Comparison with post-procest approaches
Since there are no existing post-processing approaches specifically targeting PSE, we compared the perfor- mance with other post-processing methods designed for CE. The performance was evaluated using the test accuracy and counterfactual effect (CE) on the synthetic dataset used in Section 6.2 and on the Adult dataset constructed in (Wu al., 2019a), to compare the performance with other post-process approaches. The com- parisons focus on post-process approaches, especially those for counterfactual fairness, which is special case of path-specific counterfactual fairnoss. CE was computed using the PSE equation by setting all pathways from a sensitive altribule as unfair pathways. We leveraged the Adult dataset from their websits which con- sists of 7 features extracted from the original Adult dataset, shown in the appendix. We used unfair pathways π = {S + 1,S + M- + 1,S H + 1,5 5 + M + W + S + M + H + 1, + M + W + H + 1}. where S, W, I,I denotes sex, marital-status, workelass, hours, and income, respectively. The causal graph is given in the Appendix.
We compared the proposed algorithm with algorithms suggested by (Wu et al. 2019a denoted by CF to assess its performance and used LR and SVM baseline models for post-process approaches. Similar to experiments on in-process approaches, linear regression was used as a causal model of mediators.
Figure G presents the CE and accuracy, which includes error regions averaged across 20 instances for the synthetic and Adult datasets. The runtime can be found in Figure 12 of the Appendix. The same fairness parameter settings were used for the proposed algorithms as in the experiments in Section 6.1 The fairness parameter of the CF was variod from 0 0.2 in steps of 0.01. Our algorithm with SVM outperforms other existing algorithms on the synthetic dataset, and it also achieves superior performance with both LR and SVM outperform on Adult dataset. Although our algorithm with LR the synthetic dataset is an exception, it always achieves a better CE, demonstrating its effectiveness in improving fairness.
"Other algorithms for estimating posterior distributions, such as Caussian approximation, also be applied scttings, as demonstrated (Chiapps 2019)
http://wwww.oxxxaui.co wwtp://coooppalciiiio...o
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Proposed+ +R LR
Proposed+SSM
CF+LR
CF+SVM
NN (True)
NN (Lin)
0.8
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0.65
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Figure 6: Averaged test accuracy v.s. averaged CE syn- Figure 7: Accuracy v.s. PSE on the mis- specification setting.
thetic(left) and the Adult datasots(right).
6.3 Experiment model mismatch
In Section 6.1 assume trained predictive models for mediators be true. In this section, we evaluate the performance differences between scenarios where the true causal models of mediators are known and where they are trained models. To this end, adopted true causal models mediator models when modifying YA=0, meaning that our algorithm employs the trained model instead of true model when calculating a reference value r. We then verified YĄ=1]|= using the true causal model. Figure shows the average accuracy against the PSE across 10 instances. We evaluated the proposed algorithm using neural network as a base learning model. The results for the other base models can be found Figure 13] The labels (True) and (Lin) signify that the algorithm uses the causal model and linear regression, respectively, in the computation of reference value r. These results indicate that the model mismatch had no significant impact accuracy and only slight effect PSE, up to 0.05. Furthermore, it is evident that using more precise mediator models would lead to smaller disparity.
7 Conclusion
We propose a new post-processing method to achieve path-specific counterfactual fairness. Our method is based on a set function representation of a machine learning model. By applying the Möbius inversion formula, we prove that every set function can be represented as the sum of causal effects discussed previously. To elucidate the relationship between the causal effects and the path-specific counterfactual fairness, we propose causally aware function by incorporating feasible sets and interior operator. Despite introducing the aforementioned complex concepts, the resulting algorithm is surprisingly simple and empirically works efficiently. Furthermore, our algorithm is the first post-processing method that efficiently achieves path- specific counterfactual fairness and it is model-agnostic method that can handle any model architecture and feature representation.
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Appendix
A Broader Impact Statement
Using incorrect or inaccurate causal graphs or selecting wrong unfair pathways can result in discriminatory effects, which in turn may have negative societal impacts, as highlighted by our research. A limitation of the proposed method that it ensures fairness within the context the provided causal graph, although the validity of the causal graph itself remains separate concern, beyond the scope of our method. Therefore, it crucial to continually validate and update the causal graph's validity through expert assessment. A notable advantage of the post-processing approach is that it does not require model re-training when the causal graph is updated.
B Examples
7 shall compute Aus (1A,Q, Suppose have set function with v0) = 0, v((A)) 10, v{(2)) 20, v([D)) 30, v( Q}) = 40, v((AD)) = 30, v{Q,, = 50, and v( DJ) = 100. The dividend of [A, D3 on is given by Aus (1A.Q, = ((A,)
Example 4. Having Example
o(f4, Q1) = 60 because
Lemma 5.3 can be easily understood by using linear causal models.
Example 5. In simple case where models f, SD, and fM are linear, the PSE of Figure 1 when A flips from to a', can be computed as follows. We can represent the models f, fD, and fM as fo Q) + A + +c +ed, (A)(A,Q) = c.hh ch A+c emm and [(A, Q, D, D) = c² + + where A E {a, a'), Q= Eg, and di ca dg c "",Ca cr. c² ,,, cq. cg ch td, Em, R. When set - {A + Y,A + D + Y} as unfair pathways, its PSE is given by ca aa) c2 Auy ({A)) and Aur (fA, D}) are given by Aur (1A)) ([A)) (0) fla. m') f(a',g' ,'',d'd''')) ca - d') and A (1A, D)) (x({A, DJ) (((() -- (0) I(a, dd) fa, o',,, ,d m²) aedlla a'), where (D))) = (0). Hence, the PSE can be rephrased as Doy ((A)) Auy DJ).
+
On the other hand, dividends
m²)
Example 6. Feasible sets on unfair pathwaays Fx of Example 1 is given by Fr = (0, {A}, (A,
C Detailed experimental settings
C.1 Details of baseline algorithms
PIU CChikahara et al. 2022 aims to minimize the probability of individual unfairness, which is defined by
P(YA=1]J + YA=a).
F1O (Nabi & Shpitser 2018 seeks to minimize the mean unfair effect, defined by
Ey, YYe (Y YAmo] = P(Y = 100--P(Yess P(YA=0 1).
Remove algorithm learns a predictive model using base models (logistic regression and neural network) while excluding features associated with unfair pathways.
C.2 Details of datasets comparison with in process approaches
synthetic Our synthetic data setup was based on the configuration used by (Chikahara et al., 2021] which focused on hiring decision scenario, as shown in Figure [ indicating a causal graph with X (A, D,Q, M) and A symbolizing sensitive attribute. The values of A, D, Q, and M were sampled from the
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following causal model:
A = UA: Un Bernoull(OO6)),
Q = [Vg], Uq U(0.1,1)
D A [0.5Qu]] UD (2,
M 3A + 0.4QUM, UM (3, 0001,3.0)
== Bernoulli(((110+
D MM),
where Bernoulli, 44 and TrN denotes the Bernoulli, uniform, and truncated Gaussian distributions, respec- tively, and s(x) 1/(1 exp(-x)) standard sigmoid function. We generated 10 instances, each of which is sampled 6000 data points and separated them into 5000 training data and 1000 as test data. We used {A Y, D - Y} for the synthetic dataset.
unfair pathways
Adult Our experimental setup was based on the configuration established by (Nabi & Shpitser 2018 Chikahara et al. 2021), h, for which we used the source code provided on their Github repositories 10 The adult dataset is commonly used for predicting an individual's income level by determining if their salary is above or below $50,000. Five attributes were examined: gender A, marital-status M, education L, occupation R, nationality C, from 11 attributes as shown in Figure
gender was identified as the sensitive attribute to be treated. To construct the datasets, 5000 records were selected for training and 1000 records for testing from a lotal of 65,123 records. A direct path from gender to income and indirect paths from gender to income via marital -tartal-status status to unfair pathways were established, i.e., π {A + Y.A + M + Y, - M + L + R - Y). Given unfair pathways, Punfair is given by {A, M, L, RJ.
Y,A + Y, A M
German Our german data setting was based on (Chikahara et al. 2021 Nine attributes were examined: sex A, financial information S (including saving accounts, chocking account, and housing), information about debts R (including credit amount and duration), and others C'including age, job and purpose), as sex was defined the sensitive attribute, as shown in Figure D We prepared three unfair paths from to each feature of financial information. To construct the dataset, we selected 900 records for training data and 100 records for testing from total of 1000 records. Unfair pathways are set to = {A + Y - S - Y}. Given unfair pathways, we have Punfair = [sex, including saving accounts, checking account, housing).
others
information about dehts
financial information
sex
default risk
R
Y
Figure 8: The causal graph of German setting (Chikahara al. 2022) for comparison with in-process approach. The red paths from the sensitive attribute S to Y denote unfair paths.
sensitive a astuibute
target variable
Figure 9: The causal graph of the synthetic dataset Wu et al. 2019a for comparison with post-process approach. The red paths from the sensitive attribute S to Y denote unfair paths.
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Published in Transactions on Machine Learning Research (03/2025)
education
marital-atatus
E
workelass
age
hours
Figure 10: The causal graph of Adult dataset (Wu et al. 2019a for comparison with post process approaches. The red paths from the sensitive attribute S to denote unfair paths.
C.3
Estimation of exogenous variables
We estimated the values exogenous variables through the EM algorithm. We setup the expectation and maximization steps as the following: (i) In the expectation step, each iteration, we use the current parameter estimates to compute the expected value of the exogenous variable given the observed data v), expressed as:
Z y X3 ?
where represents the estimated exogenous variable, 3 the cocfficient parameter for lincar model, and y is a noise parameter. (ii) In the maximization step, we update the parameters B and 7 as follows. We find the best fit for 3 using the least square method. As for updating y, we compute with the following formula:
7 Ev X3)2 1213
The expectation and maximization steps are iterated T times. In our experiments, for each mediator, we utilized all its observable parent attributes for and set the value of the mediator as y, and we set = 100 for the number of iterations of the EM algorithm.
C.4 Additional comparisons with in in-process approaches
Figure [I shows the test accuracy and runtime of algorithms using logistic regression, random forest, and neural networks as the base model, plotted against PSE for synthetic, Adult, and German datasets. The error bars representing the standard deviation. We varied the fairness-related parameters for each algorithm, except for the Remove algorithm. Each plot illustrates the average performance across 10 instances. The runtimes are averaged over the combinations of 10 instances and fairness parameters.
C.5 Comparisons with post-process approach
measured accuracy and counterfactual effect (CE) on the synthetic dataset used in Section 6.2 and the adult dataset used in (Wu et al. 2019a). For both the synthetic and Adult datasets, utilized dataset that was made available on their websitd The causal graph of the synthetic data is shown in Figure g Each entry of the Adult dataset. includes 7 features extracted from the original adult dataset as shown in Figure 10 We set direct path from sex to income and indirected paths from sex to income via marital-status - I,S + M + W - I,S + M + W H + I}, where S, M, W, H, denotes sex, marital-status, workclass, hours, and
are set to unfair pathways, that is, π = (S - S, > + I,S
И - I, - income, respectively. With the unfair pathway म, have Punfair {S, M, H, W). We have 65,123 data points and divided them into 80/20 training and test dataset split.
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Published in Transactions on Machine Learning Research (03/2025)
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Figure 11: Test Accuracy vs. PSE for Synthetic (Ist column), Adult (2nd column), and German datasets (3rd column), with runtime (in seconds) shown in the 4th column. Rows represent base model: Logistic Regression (1st row), Random Forest (2nd row), and Neural Network (3rd row). Error bars represent standard deviation. In the middle row, the results of PIU and FIO do not appear because they cannot use random forest as base model.
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Published in Transactions on Machine Learning Research (03/2025)
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Figure 12: Averaged test accuracy v.s. averaged CE on the synthetic(left) and Adult datasets(center) and runtimo(right).
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Figure 13:
Accuracy v.s. PSE on the misspecification setting.
Baselines & Proposed algorithm We compared proposed algorithm with algorithms proposed by (Wu et al. 2019ab denoted by CF. We utilized logistic regression (LR) and support vector machine (SVM) as baseline models of post-process approaches.
Result Figure plots the test accuracy against CE and averaged runtime when changing the parameters regarding fairness on the synthetic and the adult datasets. We used the same settings as those in the comparisons for in- process approaches in Section 6.1 as per the proposed algorithm. As for CF, we altered a fairness parameter from 0.01 to 0.3 in steps of 0.01. When compute CF, i.e., P(Ys=1 2) - P(Ys =); for both synthetic and adult dataset and for both proposed algorithms and CF, we utilized lincar regression models learned from data. lo calculate the value of each mediator in changing the sensitive altribute value to compute the first term. For each parameter setting, we averaged CE over all data points and plots them in Figure 6 and averaged runtime over all combinations of instances and fairness parameters. We can observe that in the adult dataset, our algorithms clearly outperform existing algorithms. Conversely, our algorithm with support vector machine outperforms existing algorithms, while the algorithm with logistic regression achieves lower CE but also lower accuracy than existing one. Furthermore, our algorithm significantly improve the runtime compared with the existing algorithm.
C.6
Additional experiments on model mismatch
Figure 13 shows the average accuracy against the PSE across 10 instances. We evaluated the proposed algorithm using logistic regression(L.R), random (RF)),(RF), and neural network(NN) as a base learning model. The labels (True) and (Lin) signify that the algorithm uses the true causal model and linear regression, respectively, the computation of reference value These results indicate that the model mismatch had no significant impact on accuracy and only slight effect on PSE, up to 0.05. Furthermore, it evident that using more precise mediator models would lead to smaller disparity.
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Published in Transactions on Machine Learning Research (03/2025)
D Proof of Lemma 5.1
Lemma 5.1 Let Fa be a feasible set of DAG induced by unfair pathways л. Then the prediction f' is path-spacific counterfactual fair if and only if it holds that
E SEFA(()
Aos. (S) =0
(10)
Proof. In this proof, we will write set function given a data point a and reference value on feasible set F as V.r.F. Let Ng be set of nodes excluded in unfair pathways function that considers only the effects unfair pathways x for any u. The i-th element of the reference value r' holds r = ri if on unfair pathways and r = x, otherwise, where r is a reference value constructed in Section 5.1 The resulting set function Ua,r' holds Un. " T) = (S) for all S S N and T C S N Yл=1|== and Yạ=0 can be written by using Dar"' For every S F, the dividend of U2,"" can be represented by that of vx given by
and let U2,r' be causally-aware set
D ( (5)
E (S)
Σ
Σ
(-1) (-1)--((tu Unr =(S) (TUW)
TCen(S) Wee
Σ
Σ
(-1)s-(tta
[S), WSen(S)NN
Σ
(-1)"
WCen())NN,
TCen(S)\..
(1 1))
Σ
(-1) U ()
(Sen(S)
Er 1an if (en(S) n Noel = 0
otherwise.
Va \ Noo) if n
otherwise,
where s, ft w be the size of S,T, W, respectively, the third equality follows from the property of Ux and the fifth equality follows from the binomial theorem. The equation (en(S) n N = 0 indicates that no feature in Ng is not the bottom of DAG induced by S, and the above equation shows that S \\ N₂ is a feasible set on Unr instead of S. This implies that feasible sets of 0 are equivalent to that of a DAG induced by unfair pathways n, i.c., Fr. Then we have
E (8) ( Ae (S).
SEF
SEF,
By regarding a given data point a as a reference value and the data whose sensitive attribute's value changed to as a baseline value, we have
EYa | YAeo | X = x|
Ux.r' U2,r" ((N) v2,r.((0)
Σ A , - A menn..(0 (0)
SEF
Σ A (8) - (0)
SEF,
Annnr (8).
SEF, ((0)
where the last equality follows from that it always holds 0 E F,Fz. Hence, the proof is complete.
7
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Published in Transactions on Machine Learning Research (03/2025)
E Proof of Proposition 5.3
Proposition 5.3 A dividend - (S) shows the direct causal effect if S is root node of G, the indirect causal effect if S is a pathway, and the effect of interaction if S is combined with multiple pathways.
Proof. Without loss of generality, let us consider a set of nodes denoted as N, with the top-most node being A and the other nodes can be reached from This implies that the effect of influences all other nodes within N {A].
Every feasible set conforms to of the following structural types: singleton, a pathway and the combi- nation of pathways. We prove that Ans for singleton, pathway, and the combination of pathways align with total direct effect, path-specific effect, and mediated interaction, respectively. Recall that the total effect by altering the sensitive attribute can be decomposed into the total direct effect, path-specific effects, and mediated interactions, as defined in Section 3.2
We begin by computing the direct causal effect - Y, i..., S {.). In path-specific counterfactual fairness settings, as only sensitive attribute A is intervened, we treat a direct causal effect A + Y. It evident that - -ey ({AA) represents the total direct effect, as shown by
- ({A)) us (0) - -0) ({(A))
This equation implics that - Dey ({A}) ) is the difference between the predicted value when all features changes due to the change in A and the predicted value when all features except for change. This computation is equivalent the total direct effect. For the other features E \ A, we have
(t))= c wz u 0,
where (=((i)) = (0) since is not the top-most node. In other words, i does not exist in a feasible set JF as singleton, that is, {i} & F.
Next, we show the computation about path-specific effects. Recall that the path-specific effect can be computed as the difference between the predicted value when the mediator takes on a specific value influenced by the treatment that the sensitive attribute changes from to and the predicted value when the value of mediator remain unchanged. In set function representation, every pathway S with [S| > 2 has characteristic that S has unique single node i that do not have any child, that is en(S) = {i). This indicates that node is the last mediator the pathway S. In this case, the dividend can be computed as follows.
= - us(
This equation implies that (S) is the difference between the predicted value when all features except for remain unchanged but the feature takes the value influenced by the change of the sensitive attribute. As a node works as a mediator, this computation corresponds to the path-specific effect.
Finally, we provide the computation about the mediated interaction. Recall that the mediated interaction among mediators can be computed by taking the sum of the predicted value when changing the value of mediators but their signs alternate based reprosentation, the combination of pathways, e.g., S, has characteristic that (en(S) 2 2, which can be seen as the number of mediators that do not have any child. Thus, we have
the number of changed mediators. In our set function
(S)=
(-1)
TCen(S)
This implies that the dividend can be computed as the sum of predicted value (S T), where T is the set of mediators that changes its value influenced by the alternation of the sensitive attribute and its sign differs based on the number of changed mediators T. This computation aligns with the mediated interaction.
12tn cases where the causal graph does not have single top-most node, the same problem setting can achieved by assigning the same values the nodes T' that are not influenced by A, i.e. 3e rr-
23


Reviews:
[{"review_id": "cQiJ8tHmwI", "reviewer": "Reviewer_ipuU", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n\n1. The problem studied is important, especially when auditing predictive models for fairness.\n2. The idea of dividend correction is novel and has not been previously studied in the context of path-specific counterfactual fairness.\n3. Experiments on synthetic as well as multiple real-world datasets help to assess the efficacy of the proposed approach.\n\nWeakness:\n\n1. The writing could be improved for better readability and to make the paper accessible to the audience not familiar with certain concepts introduced in the paper.\n2. Certain experimental details are not adequately discussed, such as the choice of $\\epsilon$; I have questions about this listed below. \n3. It would be helpful to discuss the approach's limitations based on the number of nodes in the causal graph.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper proposes a post-processing approach for achieving path-specific counterfactual fairness for an already developed predictive model and the availability of the causal graph used for developing the predictive model. This is achieved using tools from number theory to obtain dividends where each dividend corresponds to a causal effect along a certain pathway in the causal graph. The effect along unfair pathways is removed by removing the corresponding dividend.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "I have some questions and suggestions listed here.\n1. Does the X-axis in Figure 5 correspond to $\\epsilon$? If so, what is the reason for the range (0-0.2)?\n2. Add a legend to highlight the model represented by each color. \n3. It is unclear why the counterfactual effect (CE) was used to compare the different post-processing approaches. Removing the entire effect of the sensitive attribute may not be desirable. \n4. Figure 6 is hard to interpret, considering that the X-axis scale for the proposed approach is different from the CF methods, making it hard to understand the chosen range. Ideally, if the counterfactual effect is based only on the causal graph, then the X-axis should be the same for all the methods. I’d appreciate some clarification on this. \n5. Algorithm 1 is difficult to parse by itself; adding what $\\epsilon$ and $v(P_{unfair})$ means there would improve the readability. \n6. There’s a typo in dividends just before the contributions are listed in the introduction.\n7. In the introduction, please describe what the dividends represent. \n8. Add that the potential outcome for A set to 0 doesn’t require dependence on the unfair paths $pi$ for better readability before equation 1. \n9. Instead of saying the sensitive attribute is flipped from 0 to 1, consider stating that the sensitive attribute is intervened on to set to a particular value.\n10. In section 5, it isn't easy to parse what feasible sets, Mobius inversion formula, and dividends mean initially; I suggest improving the writing here to help the reader. \n11. I found understanding how the reference value is selected difficult. Consider adding some explanation for this.\n12. In example 1, does N represent {A,Q,D,M}?\n13. It’s not entirely clear what the synergy effect is at the end of page 7.", "broader_impact_concerns": "Since this is a post processing approach, it would be beneficial to address how to test whether the unfair paths are accurate and if they are susceptible to the location where the model is deployed.  Including a section that highlights the limitations of assuming that unfair pathways remain consistent across locations and are inherently accurate could add valuable context.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "sOaFHn1weM", "reviewer": "Reviewer_rw6V", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "**Strengths.** The proposed approach appears quite interesting and novel, as the work seems to be the first to build upon techniques on cooperative games to achieve path-specific counterfactual fairness. The authors sufficiently discuss related work and provide several examples to provide the intuition for their method. The technical results are mostly clearly presented  and the experimental results seem to include evaluation with both synthetic and real data.   \n\n**Weaknesses.** There seems to be room for improvement in terms of clarity and presentation. Some parts of the work seem as if written in a rush as there are quite a few parts that could be more clearly presented and there are several typos throughout the manuscript. In particular, the bulleted list in the introduction seems to lack flow as the first bullet highlight the efficiency of the proposed method while the second bullet actually described the method. In Section 3.1, in the description of the endogenous variables, it is unclear to what ‘quantification’ and ‘faculty type’ refer to.  In the same section, there also seems to be a typo in the definition of potential outcomes in the left term where $f_{M}(0,..)$ looks as if it should have been $f_{M}(1,..)$. In experiments, Figure 5 seems to lack a legend, therefore the reader can only speculate to which method  each line corresponds. In addition, the authors state that ‘The error bars representing the standard deviation’. However, the figures do not show error bars, but rather colored regions.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper focuses on the problem of path-specific counterfactual fairness in machine assisted decision support. In this context, the authors assume an underlying causal graph of the attributes and mediators that affect the final decision (e.g. to hire an individual), where some pathways (paths from attributes and mediators to the final decision) may be considered fair or unfair. Under this setup, the authors propose a method to adjust the output of a model predicting the final decision in order to achieve path-specific counterfactual fairness, i.e., to nullify the causal effect due to unfair pathways. To this end, the authors build upon techniques in the field of cooperative games and propose a post-training method to recompute the output of the model while removing the causal effect due to unfair mediators. For their method they propose a set function that captures parent relations between the  variables in the causal graph, which they use to compute the causal effects due to unfair pathways by decomposing it as a sum of Harsanyi dividends.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Although the presentation issues do not generally reduce the importance of the contribution, the following changes appear necessary to recommend acceptance:  \ni) A clear presentation of experimental results in Figure 5, by adding, for example, a legend indicating the compared methods \n\nii) Correcting the definition of potential outcomes in section 3.1\n\nOther changes that could improve the work include:\n\ni) Improving the flow in the contribution bullets in the introduction, by first describing the proposed method and then perhaps highlighting its efficiency. The authors may even like to remove the bullets completely and have one coherent paragraph instead.   \n\nii) Clarifying the meaning of endogenous variables in section 3.1.\n\niii) The authors discuss the selection of the reference value $r$ in case of a binary sensitive attribute. It may be worth discussing also how should one select the reference value in case of a sensitive attribute that can take more than two values from a discrete set.\n\niv) Fixing all typos \n    - Abstract: ‘We empirically show that proposed algorithm” —> ‘We empirically show that our proposed algorithm”\n\n    - Introduction, paragraph before bullet points “that effectively capture” —> “that effectively captures”\n\n    - Introduction, paragraph before bullet points: “1dividends” —> “dividends”\n\n    - Related work in the end “in the analyze” —> “in the analysis”\n\n    - Paragraph above preliminary “it also have” —> “it also has”\n\n    - Section 5.1, 1st paragraph “requires for establishing” —> “is required for” \n\n    - Below Example 2 “a cause occur” —> “a cause occurs” \n\n    - Pg 7 last line, “effect of combining with T features.”—> it is unclear with what the $T$ features are combined (perhaps “with” is not necessary).\n\n    - Bottom of page 7, “$en(T)$ is a set of nodes that do not have no child in $T$” —> Perhaps the authors meant “$en(T)$ is a set of nodes that does not have any child in $T$”\n\n    - Section 5.4, 3rd line “eahc” —> “each”\n\n    - “A notable result here is that the our post-process approach” —> “A notable result here is that our post-process approach” \n\n    - Figures 6 and 7 could have been aligned \n\n    - Section 6.3 “the other base model” —> “the other base models”\n\n    - Conclusion “we propose causally aware set function” —> “we propose a causally aware set function”", "broader_impact_concerns": "The work already includes a broader impact statement the sufficiently addresses concerns and ethical implications.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "9CJ1oTcANY", "reviewer": "Reviewer_Ntd9", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "**Strengths**\n\n1. The findings seem to present a significant contribution to the field. The proposed method seems quite non-trivial, yet the resulting algorithm is incredibly simple.\n\n2. The authors did an excellent job in terms in regards to presentation (apart from grammar mistakes, see weakness 2). Generally, the manuscript provides intuitive explanations and illustrative examples. These render the manuscript accessible to a broad audience.\n\n3. The method is agnostic of the used machine learning model.\n\n4. Many experiments on synthetic and real world data demonstrate the merits of the proposed method empirically.\n\n**Weaknesses**\n\n1. The proposed method only works for binary sensitive attributes $A$. It is easy to see that this assumption is restrictive, because many variables that lead to discrimination are not binary (e.g., ethnicity or age).\n\n2. There are quite a lot of grammar mistakes in the manuscript that are irritating. It should take little time to fix them and it will greatly enhance the reading experience.\n\n3. The theoretical preliminaries for understanding this method are quite complicated and I believe that much more effort should be spent on explaining some parts. Especially, I have a hard time understanding what this *Möbius transformer* really is and it would be great if more words could be spent on this. \n\n4. Maybe I missed it, but I could not find source code for reproducibility of the experiments.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The authors present a novel post processing method for achieving path-specific counterfactual fairness for general machine learning models. To this end, the authors leverage the Möbius inversion formula and a novel notion called *causally aware set functions* to derive a characterization for path-specific counterfactual fairness. From this characterization, the authors derive a five-step algorithm to *''correct''* any machine learning model, for which the underlying causal graph is provided. Experimental evaluations on synthetic and real world data demonstrate the merits of the proposed method in comparison to prior work.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Crucial requested changes are highlighted by the term *(crucial)*.\n\n**Abstract**\n\n- *By means of the mathematical tools in cooperative game* This seems to be a typo. I believe the authors mean something like *By means of the mathematical tools from cooperative game theory*. \n\n**Introduction** \n\n- *Their framework assumes that any causal effect of the sensitive attribute on the\noutcome as inherently problematic and aims to eliminate any influence that the sensitive attribute might\nhave on the outcome.* I do not understand this sentence. What does *problematic* mean? If we assume that no attributes should be causal for the prediction (we eliminate the influences), we end up without any attributes at all. Disclaimer: I am not very familiar with the framework by [1].\n\n- *we introduce the Möbius inversion formula* It does not seem like the authors introduce the formula. It seems instead that they introduce this formula to the field of counterfactual fairness. Please clarify.\n\n- *We newly introduce a novel causally aware set function* Typo. Please remove *newly*.\n\n**Related Work**\n\n- *n-processing approach adapts the learning algorithm...* Typo. Please write *The n-processing approach adapts the learning algorithm...*\n\n- *n-process approaches are tend to be more complex* Typo. Please remove *are*.\n\n- *Our finding will provide a new possible connection between machine learning and cooperative game* This seems to be the same typo as in the abstract. Please fix or clarify.\n\n- *in the analyze of causality.* Typo. Please write *in the analysis of causality*.\n\n**3.1 Path-specific counterfactual fairness** \n\n- *Since we focus on the impact of the sensitive attribute, we only\nconsider exogenous variables that are not influenced by the sensitive attribute.* I am a bit confused by this sentence. From my understanding, exogenous variables are never influenced by endogenous variables. Please clarify.\n\n- I do not understand why $Y_{A \\Leftarrow 0}$ does not use unfair pathways. Why is it not $Y_{A \\Leftarrow 0 || \\pi}$? The unfair mechanisms are also used here.\n\n**3.2 A decomposition of the total effect into path-specific effects**\n\n- In the definition of total effect of alternation (TE), I found it hard to see what the difference to the compared variables in the definition of path-specific counterfactual fairness (Definition 3.1) is. I believe $Y_{A \\Leftarrow 1 || \\pi}$ means that you set $A = 1$ *only* in the unfair pathways and here, you set $A = 1$ for all of pathways. It would be nice to elevate this a bit more.\n\n- *... showed that the sum of path-specific effects does not equivalent to the total effect*. Typo. Please write *... showed that the sum of path-specific effects is not equivalent to the total effect*.\n\n**4 Problem setting and motivating scenario**\n\n- *the publisher must publicly disclose the causal graph $G$ that underpins the predictive model $f$*. I am not very familiar with counterfactual fairness. However, it seems to me that most publishers simply fit their model to data without imposing a causal structure. That means they will not be able to provide such a causal graph. In this case, it seems methods such as the proposed one cannot be applied (?).\n\n- (crucial) *This is important because, when the publisher can define their causal models, the values of mediators can be adjusted as desired when altering the sensitive attribute* I do not understand this sentence. Unfortunately, it seems relevant, because it seems to be to motivation for the proposed method. Please add more explanation here, maybe using the concrete example from section 3.1.\n\n**5.1 Construction of causally aware set function**\n\n- The section title has a typo. Please write instead *5.1 Construction of causally aware set functions*.\n\n- *These conditions are necessarily...* Typo. Please replace by *These conditions are necessary*.\n\n- *must either be measured and accounted for* Typo. Please replace by *must either be measured or accounted for*.\n\n- Not a requested change, but I have a feeling that this interior operator could be related to backtracking counterfactuals ([2, 3]), though I may be wrong. The similarity seems that analagously to this operator, backtracking counterfactuals require that causal parent variables of antecedent variables are taken into account.\n\n**5.2 Decomposition into the sum of causal effects**\n\n- (crucial) Equation 6: *Intuitively, $\\Delta_{v_\\mathcal{F}}(T)$ indicates that the synergy effect of combining with T features* This sentence makes no sense (wrong grammar and no content) and I believe it would make sense to spend much more text on providing the reader with an intuition for this variable. I find this variable very hard to grasp just like that.\n\n**6.Experiments**\n\n- *and German datasets* The reference for the data set is missing.\n\n- *To this end, we utilized expectation-maximization (EM) algorithm* I do not understand why the EM algorithm is required to infer the noise variables if we assume additive noise. Specifically, $e_i = x_i - f_i(PA(x_i))$. Or are the latent variables for the EM algorithm not the noise variables, but some model parameters? If yes, it would be interesting to know what latent variable model was fit here.\n\n- *which includes error bars averaged across 20 instances for the synthetic and Adult datasets* Do these error bars correspond to 1 std.? If yes, please add that for clarity. Otherwise, please write what exactly the bars correspond to.\n\n\n[1] Kusner, Matt J., et al. \"Counterfactual fairness.\" Advances in neural information processing systems 30 (2017).\n\n[2] Von Kügelgen, Julius, Abdirisak Mohamed, and Sander Beckers. \"Backtracking counterfactuals.\" Conference on Causal Learning and Reasoning. PMLR, 2023.\n\n[3] Kladny, Klaus-Rudolf, et al. \"Deep backtracking counterfactuals for causally compliant explanations.\" Transactions on Machine Learning Research (2024).", "broader_impact_concerns": "No broader impact concerns.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: The core ideas in the submission as a novel approach path-specific fairness were perceived as interesting and strong. Throughout the discussion, the authors substantially improved the manuscript such that most questions and request from the reviewers were addressed. Overall, the submission is now in a solid state for publication at TMLR.
Decision: Accept (poster)