# Nested conformal prediction and quantile out-of-bag ensemble methods

arXiv: https://arxiv.org/abs/1910.10562 · HTML: https://mlsanity.com/paper/1910.10562/

## Key facts

- Cited by 89 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Oct 2019.

## Built on by (most cited first, 30 of 89)

- [A Gentle Introduction to Conformal Prediction and Distribution-Free Uncertainty Quantification](https://mlsanity.com/paper/2107.07511/) (2021)
- [Distribution-Free, Risk-Controlling Prediction Sets](https://mlsanity.com/paper/2101.02703/) (2021)
- [Testing for Outliers with Conformal p-values](https://mlsanity.com/paper/2104.08279/) (2021)
- [Language Models with Conformal Factuality Guarantees](https://mlsanity.com/paper/2402.10978/) (2024)
- [Distribution-free uncertainty quantification for classification under label shift](https://mlsanity.com/paper/2103.03323/) (2021)
- [Improved Online Conformal Prediction via Strongly Adaptive Online Learning](https://mlsanity.com/paper/2302.07869/) (2023)
- [Conformal Prediction using Conditional Histograms](https://mlsanity.com/paper/2105.08747/) (2021)
- [Sequential Predictive Conformal Inference for Time Series](https://mlsanity.com/paper/2212.03463/) (2022)
- [Conformalized Survival Analysis](https://mlsanity.com/paper/2103.09763/) (2021)
- [Calibrated Multiple-Output Quantile Regression with Representation Learning](https://mlsanity.com/paper/2110.00816/) (2021)
- [How to Trust Your Diffusion Model: A Convex Optimization Approach to Conformal Risk Control](https://mlsanity.com/paper/2302.03791/) (2023)
- [Improving Conditional Coverage via Orthogonal Quantile Regression](https://mlsanity.com/paper/2106.00394/) (2021)
- [PAC Prediction Sets Under Covariate Shift](https://mlsanity.com/paper/2106.09848/) (2021)
- [Decision Theoretic Foundations for Conformal Prediction: Optimal Uncertainty Quantification for Risk-Averse Agents](https://mlsanity.com/paper/2502.02561/) (2025)
- [Improving Trustworthiness of AI Disease Severity Rating in Medical Imaging with Ordinal Conformal Prediction Sets](https://mlsanity.com/paper/2207.02238/) (2022)
- [Conformal Prediction Sets with Limited False Positives](https://mlsanity.com/paper/2202.07650/) (2022)
- [Principal Uncertainty Quantification with Spatial Correlation for Image Restoration Problems](https://mlsanity.com/paper/2305.10124/) (2023)
- [One-Shot Federated Conformal Prediction](https://mlsanity.com/paper/2302.06322/) (2023)
- [Efficient and Differentiable Conformal Prediction with General Function Classes](https://mlsanity.com/paper/2202.11091/) (2022)
- [Conformal Prediction with Missing Values](https://mlsanity.com/paper/2306.02732/) (2023)
- [Probabilistically robust conformal prediction](https://mlsanity.com/paper/2307.16360/) (2023)
- [Selective Ensembles for Consistent Predictions](https://mlsanity.com/paper/2111.08230/) (2021)
- [Probabilistic Conformal Prediction with Approximate Conditional Validity](https://mlsanity.com/paper/2407.01794/) (2024)
- [Transductive conformal inference with adaptive scores](https://mlsanity.com/paper/2310.18108/) (2023)
- [Conformal Semantic Image Segmentation: Post-hoc Quantification of Predictive Uncertainty](https://mlsanity.com/paper/2405.05145/) (2024)
- [Conformal Linguistic Calibration: Trading-off between Factuality and Specificity](https://mlsanity.com/paper/2502.19110/) (2025)
- [Integrative conformal p-values for powerful out-of-distribution testing with labeled outliers](https://mlsanity.com/paper/2208.11111/) (2022)
- [Integrating Uncertainty Awareness into Conformalized Quantile Regression](https://mlsanity.com/paper/2306.08693/) (2023)
- [Learning Prediction Intervals for Regression: Generalization and Calibration](https://mlsanity.com/paper/2102.13625/) (2021)
- [Improving Uncertainty Quantification of Deep Classifiers via Neighborhood Conformal Prediction: Novel Algorithm and Theoretical Analysis](https://mlsanity.com/paper/2303.10694/) (2023)

## Abstract (arXiv)

Conformal prediction is a popular tool for providing valid prediction sets for classification and regression problems, without relying on any distributional assumptions on the data. While the traditional description of conformal prediction starts with a nonconformity score, we provide an alternate (but equivalent) view that starts with a sequence of nested sets and calibrates them to find a valid prediction set. The nested framework subsumes all nonconformity scores, including recent proposals based on quantile regression and density estimation. While these ideas were originally derived based on sample splitting, our framework seamlessly extends them to other aggregation schemes like cross-conformal, jackknife+ and out-of-bag methods. We use the framework to derive a new algorithm (QOOB, pronounced cube) that combines four ideas: quantile regression, cross-conformalization, ensemble methods and out-of-bag predictions. We develop a computationally efficient implementation of cross-conformal, that is also used by QOOB. In a detailed numerical investigation, QOOB performs either the best or close to the best on all simulated and real datasets. Code for QOOB is available at https://github.com/aigen/QOOB.