Coverage Guarantees for Pseudo-Calibrated Conformal Prediction under Distribution Shift

Prediction sets can make deployed classifiers safer by returning several plausible labels when a single prediction is uncertain. Their value depends on classwise reliability: average coverage can meet its target while rare or difficult classes fail repeatedly. This concern is sharper after distribution shift, when calibration labels come from a source environment but reliability is needed on the target. We ask what labeled source data and unlabeled target inputs reveal about class-conditional prediction sets, and when target labels are necessary. Under unrestricted joint shift, two target laws can produce the same observable data while requiring different classwise thresholds; any label-free rule covering both must enlarge its sets on one law. We give a labeled target audit that estimates the missing quantiles with a simultaneous guarantee. Probability-scale error is invariant to increasing score transformations, and threshold recovery follows under local regularity. At fixed confidence, achieving threshold tolerance $\varepsilon$ with fixed, nonadaptive class-stratified labeled pairs has total complexity $Θ(K\varepsilon^{-2}\log K)$, or $Θ(\varepsilon^{-2}\log K)$ labels per class under equal allocation. Class imbalance creates a separate acquisition cost; for foreground class probabilities of order $1/K$, the mixed-stream label complexity is also $Θ(K\varepsilon^{-2}\log K)$ at fixed confidence. Experiments on action-recognition and image shifts show that marginal coverage can conceal severe class failures and that source classwise calibration depends on the shift. The results connect the information available at deployment to the target labels needed for useful class-conditional prediction.

Conformal prediction can lose coverage when the data distribution changes after deployment. We study adaptation using labeled source data and unlabeled target inputs, allowing both the input distribution and its relationship with outcomes to change. We use Exponential Tilt Reweighting Alignment (ExTRA), introduced for classification by Maity et al. (2023), to estimate structured distribution shifts. We compare using its estimated weights in conformal calibration with additionally tilting the source predictive distribution. Shared learned predictors, estimated weights, calibration samples, and test observations isolate the effect of tilting. Existing theory gives both procedures target coverage with true weights and a common coverage bound with estimated weights. Identification calculations and an analysis of how scoring interacts with weight estimation error help explain why their performance can nevertheless differ. In a synthetic regression setting where the assumed models match the data-generating process and target inputs are informative about the shift, tilting reduces mean set length by about $30\%$ relative to weighting alone, with both methods attaining coverage near nominal. Tilting can instead cause substantial coverage losses in synthetic classification and in regression when target inputs provide little information about the response shift. Real-data experiments also show no consistent benefit. Good coverage from weighted calibration alone does not ensure that adding predictive tilting will preserve coverage. Deciding when to apply this additional adjustment using only source labels and target inputs remains an open problem.

Backward conformal prediction (BCP) fixes the size of a prediction set and estimates its miscoverage, but assumes calibration and test data are exchangeable. We extend BCP to covariate shift through importance weighting, preserving its guarantees when the density ratio is known. We show that BCP's tight miscoverage report depends only on the relative weight assigned to missed calibration examples, explaining why a better domain classifier can sometimes produce a worse estimate. Motivated by the fact that BCP depends only on nonconformity scores, we propose estimating density ratios on sorted score vectors rather than high-dimensional inputs. We prove that this approach recovers the same report asymptotically under an interpretable condition and characterize the resulting bias otherwise. Experiments with simulated and benchmark distribution shifts show that the proposed method improves BCP's miscoverage estimates, while score-based weighting consistently yields more accurate estimates than embedding-based weighting.