Unified Approach for Weakly Supervised Multicalibration

Multicalibration requires predicted scores to agree with label probabilities across rich families of subgroups and score-dependent tests, but existing methods require clean input--label pairs for evaluation and post-processing. This assumption fails in weakly supervised learning (WSL) regimes---including positive-unlabeled, unlabeled-unlabeled, and positive-confidence learning---where clean labels are costly or unavailable even though reliable uncertainty estimates may be crucial. We address this gap by developing estimators of multicalibration error and post-hoc correction methods for WSL settings in which clean input--label pairs are unavailable. We propose a unified framework for estimating and correcting multicalibration under weak supervision by combining contamination-matrix risk rewrites with witness-based calibration constraints, yielding corrected multicalibration moments with finite-sample guarantees. We further propose weak-label multicalibration boost (WLMC), a generic post-hoc recalibration algorithm under weak supervision. Finally, we conduct experiments across multiple weak-supervision settings to evaluate multicalibration behavior and deepen empirical understanding of uncertainty estimation under weak supervision.