# Backward Conformal Prediction

arXiv: https://arxiv.org/abs/2505.13732 · HTML: https://mlsanity.com/paper/2505.13732/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- Builds on 31 of its 49 references that are in the map.
- First posted on arXiv 19 May 2025.

## Builds on (most cited first, 30 of 31)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [EfficientNet: Rethinking Model Scaling for Convolutional Neural Networks](https://mlsanity.com/paper/1905.11946/) (2019)
- [XGBoost: A Scalable Tree Boosting System](https://mlsanity.com/paper/1603.02754/) (2016)
- [Classification with Valid and Adaptive Coverage](https://mlsanity.com/paper/2006.02544/) (2020)
- [Uncertainty Sets for Image Classifiers using Conformal Prediction](https://mlsanity.com/paper/2009.14193/) (2020)
- [Least Ambiguous Set-Valued Classifiers with Bounded Error Levels](https://mlsanity.com/paper/1609.00451/) (2016)
- [Conformal Prediction with Large Language Models for Multi-Choice Question Answering](https://mlsanity.com/paper/2305.18404/) (2023)
- [Learning Optimal Conformal Classifiers](https://mlsanity.com/paper/2110.09192/) (2021)
- [Safe Testing](https://mlsanity.com/paper/1906.07801/) (2019)
- [Distribution-free uncertainty quantification for classification under label shift](https://mlsanity.com/paper/2103.03323/) (2021)
- [Theoretical Foundations of Conformal Prediction](https://mlsanity.com/paper/2411.11824/) (2024)
- [Large language model validity via enhanced conformal prediction methods](https://mlsanity.com/paper/2406.09714/) (2024)
- [Hypothesis testing with e-values](https://mlsanity.com/paper/2410.23614/) (2024)
- [Prediction and outlier detection in classification problems](https://mlsanity.com/paper/1905.04396/) (2019)
- [Length Optimization in Conformal Prediction](https://mlsanity.com/paper/2406.18814/) (2024)
- [Cautious Deep Learning](https://mlsanity.com/paper/1805.09460/) (2018)
- [Provably Robust Conformal Prediction with Improved Efficiency](https://mlsanity.com/paper/2404.19651/) (2024)
- [Minimum Volume Conformal Sets for Multivariate Regression](https://mlsanity.com/paper/2503.19068/) (2025)
- [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)
- [Transductive conformal inference with adaptive scores](https://mlsanity.com/paper/2310.18108/) (2023)
- [Enhancing Conformal Prediction Using E-Test Statistics](https://mlsanity.com/paper/2403.19082/) (2024)
- [Improving Uncertainty Quantification of Deep Classifiers via Neighborhood Conformal Prediction: Novel Algorithm and Theoretical Analysis](https://mlsanity.com/paper/2303.10694/) (2023)
- [Post-selection inference for e-value based confidence intervals](https://mlsanity.com/paper/2203.12572/) (2022)
- [Beyond Neyman-Pearson: e-values enable hypothesis testing with a data-driven alpha](https://mlsanity.com/paper/2205.00901/) (2022)
- [On the Expected Size of Conformal Prediction Sets](https://mlsanity.com/paper/2306.07254/) (2023)
- [The numeraire e-variable and reverse information projection](https://mlsanity.com/paper/2402.18810/) (2024)
- [Conformal Risk Minimization with Variance Reduction](https://mlsanity.com/paper/2411.01696/) (2024)
- [Measuring Evidence against Exchangeability and Group Invariance with E-values](https://mlsanity.com/paper/2310.01153/) (2023)
- [Post-selection Inference for Conformal Prediction: Trading off Coverage for Precision](https://mlsanity.com/paper/2304.06158/) (2023)

## Built on by (most cited first, 14 of 14)

- [Conditional Coverage Diagnostics for Conformal Prediction](https://mlsanity.com/paper/2512.11779/) (2025)
- [Adaptive Coverage Policies in Conformal Prediction](https://mlsanity.com/paper/2510.04318/) (2025)
- [Calibrating Decision Robustness via Inverse Conformal Risk Control](https://mlsanity.com/paper/2510.07750/) (2025)
- [Conformal and kNN Predictive Uncertainty Quantification Algorithms in Metric Spaces](https://mlsanity.com/paper/2507.15741/) (2025)
- [Non-Asymptotic Analysis of Efficiency in Conformalized Regression](https://mlsanity.com/paper/2510.07093/) (2025)
- [E-Scores for (In)Correctness Assessment of Generative Model Outputs](https://mlsanity.com/paper/2510.25770/) (2025)
- [Beyond Procedure: Substantive Fairness in Conformal Prediction](https://mlsanity.com/paper/2602.16794/) (2026)
- [Differentially Private E-Values](https://mlsanity.com/paper/2510.18654/) (2025)
- [Improving Backward Conformal Prediction via Non-Conformity Score Transformation](https://mlsanity.com/paper/2602.01733/) (2026)
- [Beyond Fixed False Discovery Rates: Post-Hoc Conformal Selection with E-Variables](https://mlsanity.com/paper/2604.11305/) (2026)
- [Concise and Logically Consistent Conformal Sets for Neuro-Symbolic Concept-Based Models](https://mlsanity.com/paper/2605.18202/) (2026)
- [Confounding-Valid Conformal Inference for Counterfactual KPIs in Wireless Networks](https://mlsanity.com/paper/2609.05073/) (2026)
- [Set-Preserving Calibration from Conformal P-Values to E-Values](https://mlsanity.com/paper/2606.03600/) (2026)
- [Backward Conformal Prediction under Covariate Shift](https://mlsanity.com/paper/2610.16048/) (2026)

## Abstract (arXiv)

We introduce textitBackward Conformal Prediction, a method that guarantees conformal coverage while providing flexible control over the size of prediction sets. Unlike standard conformal prediction, which fixes the coverage level and allows the conformal set size to vary, our approach defines a rule that constrains how prediction set sizes behave based on the observed data, and adapts the coverage level accordingly. Our method builds on two key foundations: (i) recent results by Gauthier et al. [2025] on post-hoc validity using e-values, which ensure marginal coverage of the form mathbbP(Yrm test ∈ hat Cn^tildeα(Xrm test)) ≥ 1 - mathbbE[tildeα] up to a first-order Taylor approximation for any data-dependent miscoverage tildeα, and (ii) a novel leave-one-out estimator hatαrm LOO of the marginal miscoverage mathbbE[tildeα] based on the calibration set, ensuring that the theoretical guarantees remain computable in practice. This approach is particularly useful in applications where large prediction sets are impractical such as medical diagnosis. We provide theoretical results and empirical evidence supporting the validity of our method, demonstrating that it maintains computable coverage guarantees while ensuring interpretable, well-controlled prediction set sizes.