# Predictive inference with the jackknife+

arXiv: https://arxiv.org/abs/1905.02928 · HTML: https://mlsanity.com/paper/1905.02928/

## Key facts

- Cited by 211 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 May 2019.

## Built on by (most cited first, 30 of 211)

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- [Testing for Outliers with Conformal p-values](https://mlsanity.com/paper/2104.08279/) (2021)
- [Conformal Inference for Online Prediction with Arbitrary Distribution Shifts](https://mlsanity.com/paper/2208.08401/) (2022)
- [Learning Optimal Conformal Classifiers](https://mlsanity.com/paper/2110.09192/) (2021)
- [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)
- [Training Uncertainty-Aware Classifiers with Conformalized Deep Learning](https://mlsanity.com/paper/2205.05878/) (2022)
- [Conformalized Survival Analysis](https://mlsanity.com/paper/2103.09763/) (2021)
- [Conformal Prediction for Deep Classifier via Label Ranking](https://mlsanity.com/paper/2310.06430/) (2023)
- [Conformal Prediction for Natural Language Processing: A Survey](https://mlsanity.com/paper/2405.01976/) (2024)
- [Calibrated Multiple-Output Quantile Regression with Representation Learning](https://mlsanity.com/paper/2110.00816/) (2021)
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- [Conformal Uncertainty Sets for Robust Optimization](https://mlsanity.com/paper/2105.14957/) (2021)
- [Probabilistic Conformal Prediction Using Conditional Random Samples](https://mlsanity.com/paper/2206.06584/) (2022)
- [Length Optimization in Conformal Prediction](https://mlsanity.com/paper/2406.18814/) (2024)
- [Conformal prediction for multi-dimensional time series by ellipsoidal sets](https://mlsanity.com/paper/2403.03850/) (2024)
- [Calibrated and Sharp Uncertainties in Deep Learning via Density Estimation](https://mlsanity.com/paper/2112.07184/) (2021)
- [Conformal prediction under ambiguous ground truth](https://mlsanity.com/paper/2307.09302/) (2023)
- [Valid prediction intervals for regression problems](https://mlsanity.com/paper/2107.00363/) (2021)
- [Efficient and Differentiable Conformal Prediction with General Function Classes](https://mlsanity.com/paper/2202.11091/) (2022)
- [Conformal Validity Guarantees Exist for Any Data Distribution (and How to Find Them)](https://mlsanity.com/paper/2405.06627/) (2024)
- [Conformal Prediction with Missing Values](https://mlsanity.com/paper/2306.02732/) (2023)
- [Probabilistically robust conformal prediction](https://mlsanity.com/paper/2307.16360/) (2023)
- [Adaptive conformal classification with noisy labels](https://mlsanity.com/paper/2309.05092/) (2023)
- [Conformal Prediction: A Data Perspective](https://mlsanity.com/paper/2410.06494/) (2024)

## Abstract (arXiv)

This paper introduces the jackknife+, which is a novel method for constructing predictive confidence intervals. Whereas the jackknife outputs an interval centered at the predicted response of a test point, with the width of the interval determined by the quantiles of leave-one-out residuals, the jackknife+ also uses the leave-one-out predictions at the test point to account for the variability in the fitted regression function. Assuming exchangeable training samples, we prove that this crucial modification permits rigorous coverage guarantees regardless of the distribution of the data points, for any algorithm that treats the training points symmetrically. Such guarantees are not possible for the original jackknife and we demonstrate examples where the coverage rate may actually vanish. Our theoretical and empirical analysis reveals that the jackknife and the jackknife+ intervals achieve nearly exact coverage and have similar lengths whenever the fitting algorithm obeys some form of stability. Further, we extend the jackknife+ to K-fold cross validation and similarly establish rigorous coverage properties. Our methods are related to cross-conformal prediction proposed by Vovk [2015] and we discuss connections.