Injective Recalibration for Canonical Calibration Under Covariate Shift Without Labels

Estimating canonical calibration error (CE) on a target domain without target labels is a challenge in deployed machine learning systems facing covariate shift. We study this problem on the probability simplex, where a Dirichlet-kernel density estimator yields a consistent surrogate for the source-domain conditional ${\mathbb{E}}_S[Y\mid g(X)]$ that can be evaluated on unlabeled target predictions. Our central observation is that the resulting target CE admits a triangle-inequality upper bound into two terms: a tractable term $A(h)$ that depends on the recalibration map $h$ and is estimable from labeled source data, and a residual term $B$ that captures the source–target drift of the prediction-space label conditional. We show that injectivity of $h$ is sufficient to make the residual mismatch term independent of the choice of recalibration map within the injective family, providing a principled justification for restricting label-free recalibration to maps such as temperature scaling. We further demonstrate empirically that the injectivity of the base classifier's representation is associated with the source-to-target estimation gap of the Dirichlet-KDE estimator under rotation shift on MNIST. Finally, we introduce a label-free overlap diagnostic that detects regions of the target prediction distribution lying outside the source support, where absolute continuity fails and the surrogate becomes unreliable. Across MNIST and SVHN under strict reweighting-induced covariate shift, CIFAR-10-C, CIFAR-10$\to$CINIC-10, Camelyon17-WILDS, and OfficeHome, the proposed method matches or improves upon SOTA methods on most settings; we additionally characterize the regime—severe shift with collapsed support overlap—where all source-only methods, including ours, fail, and argue that the overlap diagnostic should be reported alongside any such CE estimate.