# Distribution-Free Predictive Inference For Regression

arXiv: https://arxiv.org/abs/1604.04173 · HTML: https://mlsanity.com/paper/1604.04173/

## Key facts

- Cited by 673 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Apr 2016.

## Built on by (most cited first, 30 of 673)

- [A Gentle Introduction to Conformal Prediction and Distribution-Free Uncertainty Quantification](https://mlsanity.com/paper/2107.07511/) (2021)
- [Adaptive Conformal Inference Under Distribution Shift](https://mlsanity.com/paper/2106.00170/) (2021)
- [Conformal Risk Control](https://mlsanity.com/paper/2208.02814/) (2022)
- [Conformal Language Modeling](https://mlsanity.com/paper/2306.10193/) (2023)
- [Adaptive Conformal Predictions for Time Series](https://mlsanity.com/paper/2202.07282/) (2022)
- [Human-Centered Explainable AI (XAI): From Algorithms to User Experiences](https://mlsanity.com/paper/2110.10790/) (2021)
- [Class-Conditional Conformal Prediction with Many Classes](https://mlsanity.com/paper/2306.09335/) (2023)
- [Distribution-free uncertainty quantification for classification under label shift](https://mlsanity.com/paper/2103.03323/) (2021)
- [C-TPT: Calibrated Test-Time Prompt Tuning for Vision-Language Models via Text Feature Dispersion](https://mlsanity.com/paper/2403.14119/) (2024)
- [Conformal Prediction using Conditional Histograms](https://mlsanity.com/paper/2105.08747/) (2021)
- [Selection by Prediction with Conformal p-values](https://mlsanity.com/paper/2210.01408/) (2022)
- [Batch Multivalid Conformal Prediction](https://mlsanity.com/paper/2209.15145/) (2022)
- [Practical Adversarial Multivalid Conformal Prediction](https://mlsanity.com/paper/2206.01067/) (2022)
- [Training Uncertainty-Aware Classifiers with Conformalized Deep Learning](https://mlsanity.com/paper/2205.05878/) (2022)
- [Conformal Alignment: Knowing When to Trust Foundation Models with Guarantees](https://mlsanity.com/paper/2405.10301/) (2024)
- [Conformal Prediction Under Feedback Covariate Shift for Biomolecular Design](https://mlsanity.com/paper/2202.03613/) (2022)
- [Conformal Prediction for Natural Language Processing: A Survey](https://mlsanity.com/paper/2405.01976/) (2024)
- [Few-shot Conformal Prediction with Auxiliary Tasks](https://mlsanity.com/paper/2102.08898/) (2021)
- [Conformalized Survival Analysis](https://mlsanity.com/paper/2103.09763/) (2021)
- [Online Multivalid Learning: Means, Moments, and Prediction Intervals](https://mlsanity.com/paper/2101.01739/) (2021)
- [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)
- [Federated Conformal Predictors for Distributed Uncertainty Quantification](https://mlsanity.com/paper/2305.17564/) (2023)
- [Decision Theoretic Foundations for Conformal Prediction: Optimal Uncertainty Quantification for Risk-Averse Agents](https://mlsanity.com/paper/2502.02561/) (2025)
- [Achieving Risk Control in Online Learning Settings](https://mlsanity.com/paper/2205.09095/) (2022)
- [Length Optimization in Conformal Prediction](https://mlsanity.com/paper/2406.18814/) (2024)
- [Explaining Predictive Uncertainty with Information Theoretic Shapley Values](https://mlsanity.com/paper/2306.05724/) (2023)
- [On the Utility of Prediction Sets in Human-AI Teams](https://mlsanity.com/paper/2205.01411/) (2022)
- [Predictive Inference with Feature Conformal Prediction](https://mlsanity.com/paper/2210.00173/) (2022)
- [Object Pose Estimation with Statistical Guarantees: Conformal Keypoint Detection and Geometric Uncertainty Propagation](https://mlsanity.com/paper/2303.12246/) (2023)

## Abstract (arXiv)

We develop a general framework for distribution-free predictive inference in regression, using conformal inference. The proposed methodology allows for the construction of a prediction band for the response variable using any estimator of the regression function. The resulting prediction band preserves the consistency properties of the original estimator under standard assumptions, while guaranteeing finite-sample marginal coverage even when these assumptions do not hold. We analyze and compare, both empirically and theoretically, the two major variants of our conformal framework: full conformal inference and split conformal inference, along with a related jackknife method. These methods offer different tradeoffs between statistical accuracy (length of resulting prediction intervals) and computational efficiency. As extensions, we develop a method for constructing valid in-sample prediction intervals called it rank-one-out conformal inference, which has essentially the same computational efficiency as split conformal inference. We also describe an extension of our procedures for producing prediction bands with locally varying length, in order to adapt to heteroskedascity in the data. Finally, we propose a model-free notion of variable importance, called it leave-one-covariate-out or LOCO inference. Accompanying this paper is an R package tt conformalInference that implements all of the proposals we have introduced. In the spirit of reproducibility, all of our empirical results can also be easily (re)generated using this package.