Decomposing Direct and Indirect Biases in Linear Models under Demographic Parity Constraint
Bertille Tierny, Arthur Charpentier, François Hu
TL;DR
The paper addresses fairness in linear models under Demographic Parity by developing a post-processing framework that decomposes predictive bias into direct and indirect sources and derives a closed-form optimal fair predictor $f^*_$. It provides a Gaussian linear model that captures direct bias, indirect mean bias and indirect structural bias, and shows how to interpolate between the Bayes predictor and a DP fair predictor to navigate the risk fairness frontier. A complete bias decomposition is presented at both the prediction and feature levels, including a tractable additive approximation under uncorrelated features and a general interaction based extension. A plug-in estimator enables practical estimation from finite data, and experiments on synthetic and real data demonstrate improved fairness with interpretable coefficient adjustments and competitive accuracy, enabling auditing and mitigation without retraining.
Abstract
Linear models are widely used in high-stakes decision-making due to their simplicity and interpretability. Yet when fairness constraints such as demographic parity are introduced, their effects on model coefficients, and thus on how predictive bias is distributed across features, remain opaque. Existing approaches on linear models often rely on strong and unrealistic assumptions, or overlook the explicit role of the sensitive attribute, limiting their practical utility for fairness assessment. We extend the work of (Chzhen and Schreuder, 2022) and (Fukuchi and Sakuma, 2023) by proposing a post-processing framework that can be applied on top of any linear model to decompose the resulting bias into direct (sensitive-attribute) and indirect (correlated-features) components. Our method analytically characterizes how demographic parity reshapes each model coefficient, including those of both sensitive and non-sensitive features. This enables a transparent, feature-level interpretation of fairness interventions and reveals how bias may persist or shift through correlated variables. Our framework requires no retraining and provides actionable insights for model auditing and mitigation. Experiments on both synthetic and real-world datasets demonstrate that our method captures fairness dynamics missed by prior work, offering a practical and interpretable tool for responsible deployment of linear models.