Fairness Under Group-Conditional Prior Probability Shift: Invariance, Drift, and Target-Aware Post-Processing

Machine learning systems are often trained and evaluated for fairness on historical data, yet deployed in environments where conditions have shifted. A particularly common form of shift occurs when the prevalence of positive outcomes changes differently across demographic groups--for example, when disease rates rise faster in one population than another, or when economic conditions affect loan default rates unequally. We study group-conditional prior probability shift (GPPS), where the label prevalence $P(Y=1\mid A=a)$ may change between training and deployment while the feature-generation process $P(X\mid Y,A)$ remains stable. Our analysis yields three main contributions. First, we prove a fundamental dichotomy: fairness criteria based on error rates (equalized odds) are structurally invariant under GPPS, while acceptance-rate criteria (demographic parity) can drift--and we prove this drift is unavoidable for non-trivial classifiers (shift-robust impossibility). Second, we show that target-domain risk and fairness metrics are identifiable without target labels: the invariance of ROC quantities under GPPS enables consistent estimation from source labels and unlabeled target data alone, with finite-sample guarantees. Third, we propose TAP-GPPS, a label-free post-processing algorithm that estimates prevalences from unlabeled data, corrects posteriors, and selects thresholds to satisfy demographic parity in the target domain. Experiments validate our theoretical predictions and demonstrate that TAP-GPPS achieves target fairness with minimal utility loss.