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PC-Fairness: A Unified Framework for Measuring Causality-based Fairness
Yongkai Wu, Lu Zhang, Xintao Wu, Hanghang Tong
TL;DR
Causality-based fairness can be hard to measure when its underlying causal effects are not identifiable from observational data. This paper unifies many such fairness notions as PC fairness and bounds it using constrained optimization. Experiments report bounds for unidentifiable settings and tighter bounds than previous methods.
Problem
Many causal effects underlying causality-based fairness cannot be uniquely measured from observational data, limiting identifiability.
Source: §1 · p. 1Method
The paper defines PC fairness using path-specific counterfactual effects and bounds it through constrained optimization over parameterized causal models.
Source: §3 · p. 4, §4 · p. 5, §4.1 · p. 5Results
Across synthetic and real-world datasets, the method bounds causal effects under unidentifiable situations and combinations and achieves tighter bounds than previous methods.
Source: §6 · p. 9Takeaways & Limitations
PC fairness can define and analyze individual indirect discrimination, which previous fairness notions had not studied.
Source: §3 · p. 4The response-variable formulation can scale exponentially with total graph in-degree, though variables directly causing unidentification may suffice.
Source: §6 · p. 9
Abstract
from arXiv · showhide
A recent trend of fair machine learning is to define fairness as causality-based notions which concern the causal connection between protected attributes and decisions. However, one common challenge of all causality-based fairness notions is identifiability, i.e., whether they can be uniquely measured from observational data, which is a critical barrier to applying these notions to real-world situations. In this paper, we develop a framework for measuring different causality-based fairness. We propose a unified definition that covers most of previous causality-based fairness notions, namely the path-specific counterfactual fairness (PC fairness). Based on that, we propose a general method in the form of a constrained optimization problem for bounding the path-specific counterfactual fairness under all unidentifiable situations. Experiments on synthetic and real-world datasets show the correctness and effectiveness of our method.
1 Introduction
Causality-based fairness is difficult to apply when its causal effects cannot be uniquely identified from observational data. The paper unifies existing fairness notions and develops a bounding framework for these unidentifiable settings.
1 Introduction
- Identifiability is a central barrier because many causal effects cannot be uniquely measured from observational data, while simplifying assumptions may damage predictive-model performance.Prior bounding methods address indirect discrimination and counterfactual fairness, but their tightness and applicability to combinations of unidentifiable situations were unclear.Source: §1 · p. 1
- The paper defines path-specific counterfactual fairness (PC fairness) to unify most previous causality-based fairness notions.It represents causal effects using path-specific counterfactual effects and develops a constrained optimization method to bound PC fairness across unidentifiable situations and their combinations.Source: §1 · p. 2
- Experiments on synthetic and real-world datasets show that the method bounds causal effects under unidentifiable situations and combinations, with tighter bounds than prior methods for path-specific or counterfactual effects.The framework assumes the causal graph is given, but makes no assumptions about hidden confounders or whether data-generating functions are linear or nonlinear.Source: §1 · p. 2
- Compared with PSCF, the paper extends path-specific counterfactual fairness beyond individuals to subgroups and addresses identifiability under individual-level conditioning.It distinguishes this effect from the conditional path-specific effect in: the former conditions on a post-intervention distribution, while the latter conditions on a pre-intervention distribution.Source: §1 · p. 2
2 Preliminaries
The preliminaries establish notation and introduce causal models, graphs, and three kinds of causal effects. These concepts provide the basis for describing fairness through causal pathways and factual observations.
2 Preliminaries
- Uppercase letters denote variables, bold uppercase letters denote sets of variables, and lowercase letters denote values or sets of values.Source: §2 · p. 2
2.1 Causal Model and Causal Graph
- A structural causal model specifies exogenous and endogenous variables, their equations, and a joint distribution over exogenous variables; Markovian models have independent exogenous variables, while semi-Markovian models need not.Causal graphs represent variable dependencies; hidden common causes in semi-Markovian models are shown by bidirected edges between their children.Source: §2.1 · pp. 2–3
2.2 Causal Effects
- The total causal effect transmits an intervention along all causal paths, whereas a path-specific effect transmits it only along a selected path set and uses a reference intervention on other paths.Source: §2.2 · p. 3
- A counterfactual effect conditions inference on factual observations O=o, restricting the causal inference to the subpopulation specified by those observations.Source: §2.2 · p. 3
3 Path-specific Counterfactual Fairness
The section defines PC fairness by requiring a path-specific counterfactual effect to be zero under a factual condition and chosen paths. This parameterization recasts prior causality-based fairness notions as special cases and accommodates individual indirect discrimination.
3 Path-specific Counterfactual Fairness
- The path-specific counterfactual effect measures the effect of changing X from x0 to x1 on Y through paths π, given O=o and reference x0.Source: §3 · p. 4
- In the fair-ML setting, S is protected, X contains non-protected attributes, and a predictor maps X and S to the decision Ŷ.Π denotes all causal paths from S to Ŷ in the causal graph.Source: §3 · p. 4
- PC fairness holds when the path-specific counterfactual effect is zero for a predictor under a factual condition and chosen causal paths.The condition is defined for O ⊆ {S, X, Y} and a path set π.Source: §3 · p. 4
- Prior fairness notions are special cases of PC fairness, categorized by whether O is empty and whether π represents total, direct, or indirect effects.The direct-path set is πd; πi contains paths through redlining attributes.Source: §3 · p. 4
- PC fairness can define and analyze individual indirect discrimination by setting O={S, X} and π=πi, a case not previously studied in the literature.The passage attributes this gap probably to difficulty in definition and identification.Source: §3 · p. 4
4 Measuring Path-specific Counterfactual Fairness
The section develops a general method for bounding path-specific counterfactual effects in unidentifiable situations. It represents causal models through response-function variables and uses observational consistency constraints to find tight bounds.
4 Measuring Path-specific Counterfactual Fairness
- Bow, kite, and w-graph patterns capture different unidentifiability cases; the method bounds the path-specific counterfactual effect by optimizing over observationally compatible causal models.The bow graph is a simplest hedge case; kite and w graphs correspond to additional conditions on O and π.Source: §4 · pp. 4–5
4.1 Response-function Variable
- Response-function variables encode each endogenous variable’s deterministic response to its parents, reducing structural functions to a finite set of response types.Their finite domains permit the observational joint distribution to be expressed linearly in the distribution over response-function variables.Source: §4.1 · pp. 5–6
4.2 Expressing Path-specific Counterfactual Fairness
- The path-specific counterfactual expression enforces factual observations and assigns different interventions to witness, on-path, and off-path variables.Witness variables take separate values under π and its complement, while A and B denote non-witness variables on π and its complement, respectively.Source: §4.2 · pp. 6–7
4.3 Bounding Path-specific Counterfactual Fairness
- Minimizing or maximizing the path-specific counterfactual effect over response-function distributions consistent with observed data yields tight lower and upper bounds.For τ-PC fairness, the bounds can establish satisfaction, establish non-satisfaction, or leave the status uncertain.Source: §4.3 · p. 8
5 Experiments
Experiments validate the bounds against synthetic ground truth and compare their tightness with prior methods on D2 and Adult.
5 Experiments
- Evaluation used synthetic datasets from two causal-model versions and the Adult dataset, which contains 65,123 records.The synthetic versions vary whether exogenous variables share a hidden confounder or are mutually independent.Source: §5 · p. 8
- On D1, the true path-specific counterfactual effect falls within the proposed bounds for every value combination of O.The ground truth is computed by executing interventions in the complete causal model.Source: §5 · p. 8, Table 2 · p. 9
- On Adult, the method’s bounds for CE(s+, s−|o) yield results similar to.The comparison uses O = {age, edu, marital-status}.Source: §5 · p. 9, Table 4 · p. 9
- 0.1772 exceeds τ = 0.1, so the method guarantees discrimination where ’s [−0.2605, 0.2656] bounds are uncertain.The comparison is on D2 and concerns indirect discrimination.Source: §5 · pp. 8–9, Table 3 · p. 9
- For the 2nd O groups, ’s [−0.4383, −0.0212] bounds are uncertain, while [−0.0783, −0.0212] guarantees fairness.This comparison measures counterfactual fairness on D2.Source: Table 3 · p. 9, §5 · p. 9
6 Conclusion
The conclusion presents PC fairness bounds as a unified approach to causal fairness measurement, while noting formulation scalability and fair-model construction as open concerns.
6 Conclusion
- The framework defines PC fairness to cover most prior causality-based fairness notions and formulates linear programming bounds claimed to be tightest possible.Experiments are reported to bound causal effects under any unidentifiable situation or combination and to achieve tighter bounds than previous methods.Source: §6 · p. 9
- Response-variable domains grow exponentially with parent count, making formulation complexity potentially large for high-in-degree causal graphs.The authors note that using only response variables that directly lead to unidentification could significantly decrease total complexity.Source: §6 · p. 9
- Constructing fair predictive models from the derived bounds remains future work.One proposed direction incorporates the bounding formulation into post-processing as a min-max optimization problem.Source: §6 · p. 9