Tuning Derivatives for Causal Fairness in Machine Learning

Jul 16, 2026

Artificial-intelligence systems increasingly shape decisions, but their predictions often reproduce bias tied to protected attributes such as race, gender, and age. Classical Statistical Parity (SP) enforces independence from protected attributes, yet can be too restrictive when these attributes affect mediators that are treated as legitimate business necessities. The paper presents a causal framework for structural causal models, tailored to continuous protected attributes, formalizing SP and Predictive Parity (PP) via path-specific partial derivatives, analyzing when fair predictors exist, and proposing a fair tuning algorithm with an SP–PP trade-off when necessary. Experiments on simulated and real data show improved performance when PP is prioritized.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages32

Reading Time1 h 21 min

Introduction

Fairness notions and limitations of Statistical Parity

Causal formulation using Structural Causal Models

Framework and criteria for causal fairness

Formalizing SP and PP with path-specific partial derivatives

Existence of fair predictors and conditions

Fair tuning algorithm

Constructing fair predictors

Handling infeasible cases and SP–PP trade-offs

Experiments and results

Evaluation on simulated and real data

Comparisons with prior methods

arXiv :2605 .05882v2 [ stat .ML] 10 Jul 2026 Tuning Derivatives for Causal Fairness in Machine Learning Filip Edstr¨om 1*, Guilherme W. F. Barros2 , Tetiana Gorbach 1 , Xavier de Luna 1 1* Department of Statistics, Ume˚a School of Business, Economics and Statistics, Ume˚a University, Ume˚a, 901 87, Sweden. 2 Integrated Science Lab, Department of Physics, Ume˚a University, Ume˚a, 901 87, Sweden. *Corresponding author(s). E-mail(s): [filip.edstrom@umu.se](filip.edstrom@umu.se) ; Contributing authors: [guilherme.barros@umu.se](guilherme.barros@umu.se) ; [tetiana.gorbach@umu.se](tetiana.gorbach@umu.se) ; [xavier.de.luna@umu.se](xavier.de.luna@umu.se) ; Abstract Artificial-intelligence systems are becoming ubiquitous in society, yet their predictions typically inherit biases with respect to protected attributes such as race, gender, or age. Classical fairness notions, most notably Statistical Parity (SP), demand that predictions be independent of the protected attributes, but are overly restrictive when these attributes influence mediating variables that are considered business necessities. Recent causal formulations relax SP by distinguishing allowed from not-allowed causal paths and by complementing SP with Predictive Parity (PP), requiring the predictor to replicate the legitimate influence of business-necessities. Existing path-based definitions are mainly practical when applied to categorical attributes. This paper introduces a new framework for fairness in structural causal models that is tailored to continuous protected attributes. We formalize SP and PP through path-specific partial derivatives, establish conditions under which these criteria coincide with prior causal definitions, and characterize when a fair predictor, one that satisfies SP along not-allowed paths while achieving PP along allowed paths, exists. Building on this theory, we propose a fair tuning algorithm that either constructs such a predictor or, when not possible, allows for a trade-off between SP and PP. We present experiments on simulated and real data to evaluate our proposal, compare it with previously proposed methods, and show that it performs better when PP is considered. Published in Machine Learning (Springer) on May 18, 2026 DOI: 10.1007/s10994-026-07061-7 Keywords: Structural Causal Models, Path-Specific Effects, Causal Fairness, Statistical Parity, Predictive Parity 1 Introduction Artificial-Intelligence (AI) systems are increasingly being incorporated into decisionmaking processes. Yet, AI systems have been criticized for replicating discriminatory behaviors observed in society, for example, racial bias in the COMPAS recidivism risk assessment, gender bias in Amazon’s hiring system, and bias with respect to age (Stypinska, 2023) . 1 2 When regulation or ethical aspects do not allow replicating bias with respect to a feature, for example, age, then we call such a feature a “protected attribute”. Fairness in machine learning is the field devoted to defining metrics to evaluate biases with respect to protected attributes and to finding ways to develop AI systems that avoid or minimize biases existing in the data used to create these systems. obtConsiainedduesrisnitguationsdata forwhwheircehaodanetaor-drivseveenralpredictorprotecteaforttria varbutesiabXlearoefrintelaetreesdttYoYis Several definitions of what constitutes fair predictors have been proposed, see Barocas, Hardt, and Narayanan (2023) for an overview. An early and intuitive fairness condition, Statistical Parity (SP) (Darlington, 1971), requires that predictions be indepsterndictenifttof protected athe protected atttrriibbututeess,dthatepenisdo, Xher. Howevattribuetre,s tthhisat conditionare deememayd necbeesstaoroy for prediction, called “business necessities” in Pleˇcko and Bareinboim (2024b) and“resolving variables” in Kilbertus et al. (2017) . As an example, consider a recidivism risthkeassessment sprobability ofysrteemcidi,vsuchismfaosr CanOoMPASff

FAQs

What problem does the paper address in machine learning fairness?

The paper addresses how AI predictions can inherit biases related to protected attributes like race, gender, and age. It focuses on designing fair predictors when protected attributes influence intermediate variables that may be considered business necessities.

How do Statistical Parity (SP) and Predictive Parity (PP) differ in this work?

SP requires predictions to be independent of protected attributes, but that can be overly restrictive when protected attributes affect allowed mediating variables. PP complements SP by requiring the predictor to replicate the legitimate influence of business-necessities, implemented through path-specific constraints.

What is the core contribution of the proposed framework?

The paper introduces a fairness framework for structural causal models tailored to continuous protected attributes. It formalizes SP and PP using path-specific partial derivatives, derives conditions for equivalence with prior causal definitions, and characterizes when a fair predictor exists.

Jul 16, 2026

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