# Conformal Bayes under Label Shift: Post-Hoc Calibration vs. In-Training Adaptation

arXiv: https://arxiv.org/abs/2606.11865 · HTML: https://mlsanity.com/paper/2606.11865/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 24 references that are in the map.
- First posted on arXiv 10 Jun 2026.

## Builds on (most cited first, 12 of 12)

- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [A Gentle Introduction to Conformal Prediction and Distribution-Free Uncertainty Quantification](https://mlsanity.com/paper/2107.07511/) (2021)
- [Distribution-Free Predictive Inference For Regression](https://mlsanity.com/paper/1604.04173/) (2016)
- [Conformalized Quantile Regression](https://mlsanity.com/paper/1905.03222/) (2019)
- [Weight Uncertainty in Neural Networks](https://mlsanity.com/paper/1505.05424/) (2015)
- [Conformal Prediction Under Covariate Shift](https://mlsanity.com/paper/1904.06019/) (2019)
- [Detecting and Correcting for Label Shift with Black Box Predictors](https://mlsanity.com/paper/1802.03916/) (2018)
- [Conformal prediction beyond exchangeability](https://mlsanity.com/paper/2202.13415/) (2022)
- [Regularized Learning for Domain Adaptation under Label Shifts](https://mlsanity.com/paper/1903.09734/) (2019)
- [Distribution-free uncertainty quantification for classification under label shift](https://mlsanity.com/paper/2103.03323/) (2021)
- [Conformal Bayesian Computation](https://mlsanity.com/paper/2106.06137/) (2021)
- [Uncertainty Quantification for Deep Regression using Contextualised Normalizing Flows](https://mlsanity.com/paper/2512.00835/) (2025)

## Built on by (most cited first, 5 of 5)

- [Conformal Bayes for Two-Sided Censored Gaussian Regression under Label Shift](https://mlsanity.com/paper/2607.02173/) (2026)
- [Anytime-Valid Confirmation of Label-Shift Corrections](https://mlsanity.com/paper/2606.14028/) (2026)
- [Null-Calibrated Conformal Selection via Target-Membership Scores](https://mlsanity.com/paper/2606.22336/) (2026)
- [Conformal Bayes under Continuous Label Shift: Sensitivity Analysis and the Limits of Exact Validity](https://mlsanity.com/paper/2609.27976/) (2026)
- [Conformal Prediction under Exponential-Tilt Joint Shift](https://mlsanity.com/paper/2609.30886/) (2026)

## Abstract (arXiv)

Conformal Bayes combines Bayesian posterior predictives with conformal calibration to produce prediction sets that are both statistically valid and geometrically efficient. We study conformal Bayes under label shift from a unified perspective, identifying two complementary approaches that restore nominal target-domain coverage through importance-weighted conformal calibration but operate through independent mechanisms. Post-hoc calibration tilts the posterior predictive toward the target domain and corrects the conformal threshold via an importance-weighted quantile, leaving the parameter posterior unchanged. In-training adaptation tilts the parameter posterior itself to the target domain, producing a corrected predictive whose highest predictive density region serves as the highest predictive density (HPD)-based prediction set under the fitted target predictive; efficiency is model-dependent and does not imply finite-sample conditional optimality. Two controlled experiments isolate the regime-dependence of each strategy: in the low-dimensional, well-estimated regime Strategy~A produces the narrowest valid intervals, while in the high-dimensional, underdetermined regime Strategy~B achieves up to 43% width reduction at unchanged coverage, under the stated source-sampling and label-shift assumptions.