Conformal Bayes under Continuous Label Shift: Sensitivity Analysis and the Limits of Exact Validity

Abstract

Conformal Bayes combines Bayesian posterior predictive scores with conformal calibration, but under continuous label shift both the score and calibration weight depend on the unknown response-marginal density ratio. Existing methods typically estimate one shift parameter from pseudo-labels or predictive samples and plug it into calibration. We instead propose Joint Tilt-Sensitivity Conformal Bayes (JTS-CB), which performs sensitivity analysis over a prespecified set of plausible tilts; its split-conformal realization is JTS-SCB. Each tilt jointly determines the Bayesian conformal score and conformal importance weight. JTS-SCB forms a bounded sensitivity envelope over candidate tilts, but its calibration-only construction does not inherit the exact finite-sample weighted-conformal guarantee. We therefore study a separate candidate-weighted exact counterpart and show that its usefulness depends sharply on tail behavior. For scalar linear exponential tilts, any nonzero candidate tilt makes the exact set unbounded. More generally, tail-growing density ratios produce the same pathology, whereas quadratic tilts with a negative coefficient on \(y^2\) have vanishing tail weights and admit bounded exact inference on the original target. Ratio clipping provides a complementary bounded exact construction for a surrogate target when tails grow. Experiments show that strong plug-in predictive sampling can match the oracle when the shift is well identified, while sensitivity analysis is most useful for richer, weakly identified, or systematically biased shift models, at the cost of wider prediction sets.