# Backward Conformal Prediction under Covariate Shift

arXiv: https://arxiv.org/abs/2610.16048 · HTML: https://mlsanity.com/paper/2610.16048/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 18 of its 29 references that are in the map.
- First posted on arXiv 12 Oct 2026.

## Builds on (most cited first, 18 of 18)

- [Wide Residual Networks](https://mlsanity.com/paper/1605.07146/) (2016)
- [Benchmarking Neural Network Robustness to Common Corruptions and Perturbations](https://mlsanity.com/paper/1903.12261/) (2019)
- [AugMix: A Simple Data Processing Method to Improve Robustness and Uncertainty](https://mlsanity.com/paper/1912.02781/) (2019)
- [RobustBench: a standardized adversarial robustness benchmark](https://mlsanity.com/paper/2010.09670/) (2020)
- [Conformal Prediction Under Covariate Shift](https://mlsanity.com/paper/1904.06019/) (2019)
- [Conformal prediction beyond exchangeability](https://mlsanity.com/paper/2202.13415/) (2022)
- [Do CIFAR-10 Classifiers Generalize to CIFAR-10?](https://mlsanity.com/paper/1806.00451/) (2018)
- [Leveraging Unlabeled Data to Predict Out-of-Distribution Performance](https://mlsanity.com/paper/2201.04234/) (2022)
- [PAC Prediction Sets Under Covariate Shift](https://mlsanity.com/paper/2106.09848/) (2021)
- [Model-free selective inference under covariate shift via weighted conformal p-values](https://mlsanity.com/paper/2307.09291/) (2023)
- [E-Values Expand the Scope of Conformal Prediction](https://mlsanity.com/paper/2503.13050/) (2025)
- [Conformal e-prediction](https://mlsanity.com/paper/2001.05989/) (2020)
- [Enhancing Conformal Prediction Using E-Test Statistics](https://mlsanity.com/paper/2403.19082/) (2024)
- [WATCH: Adaptive Monitoring for AI Deployments via Weighted-Conformal Martingales](https://mlsanity.com/paper/2505.04608/) (2025)
- [Backward Conformal Prediction](https://mlsanity.com/paper/2505.13732/) (2025)
- [Conformal Selective Prediction with General Risk Control](https://mlsanity.com/paper/2603.24704/) (2026)
- [Boosting e-BH via conditional calibration](https://mlsanity.com/paper/2404.17562/) (2024)
- [Improving Backward Conformal Prediction via Non-Conformity Score Transformation](https://mlsanity.com/paper/2602.01733/) (2026)

## Abstract (arXiv)

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.