Conformal Bayes for Two-Sided Censored Gaussian Regression under Label Shift

Prediction under label shift becomes nonstandard when responses are censored.

In a two-sided censored Gaussian model, latent values below $L$ and above $U$ are recorded at the boundary values, so the observed predictive distribution is mixed, with atoms at $L$ and $U$ and a continuous density on $(L,U)$.

In this paper we develop conformal Bayes for this mixed-space setting by combining posterior predictive tilting with weighted conformal calibration.

Under a two-sided Tobit Gaussian Bayesian prediction head with a Laplace posterior approximation, the tilted predictive distribution has left-atom, interior, and right-atom components, with a three-term closed-form normalizer.

The resulting prediction set is a mixed highest density region that can combine boundary atoms with an interior interval and can reduce to atom-only sets under strong censoring.

The main technical issue is that latent label shift does not directly give an ordinary density ratio on the observed censored scale.

A latent exponential tilt induces tail-averaged atom weights at the censored boundaries, while the interior ratio remains density based.

This yields a mixed observed-space calibration weight with two atom ratios and one interior density ratio.

The weight corrects the calibration measure, while predictive tilting gives target-adapted mixed-HDR geometry.

Synthetic experiments show that weighted tilted conformal Bayes restores marginal coverage with smaller sets than weighted source-score calibration, while revealing a trade-off between marginal coverage and component-wise behavior across atoms and interior observations.