Distribution-Free Predictive Inference for Regression
Open accessAbstract
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 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, to adapt to heteroscedasticity in the data. Finally, we propose a model-free notion of variable importance, called leave-one-covariate-out or LOCO inference. Accompanying this article is an R package 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.
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The risk lower bound in Theorem 2 has a slightly worse constant than the usual conformal guarantee; it recovers the conformal lower bound P(Y_{n+1} ∉ C_{λ̂}(X_{n+1})) ≥ α - 1/(n+1) as P(Y_{n+1} ∉ C_{λ̂}(X_{n+1})) ≥ α - 2/(n+1), a factor of two worse.
“However, the risk lower bound in Theorem 2 has a slightly worse constant than the usual conformal guarantee.”
From Conformal Risk Control (2022) · cites this paper as [8]
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Self-citationInductive Confidence Machines for Regression
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Peeking Inside the Black-Box: A Survey on Explainable Artificial Intelligence (XAI)
Adadi, A., & Berrada, M. (2018) · IEEE Access · doi:10.1109/access.2018.2870052 · Cited by 6,294 · see citing works
“LOCO [86] use a local feature importance as well. d: EXAMPLE-BASED EXPLANATION Example-based explanation techniques select particular instances of the dataset to explain the behavior of machine learning models.”
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Interpretable Machine Learning – A Brief History, State-of-the-Art and Challenges
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Conformalized Quantile Regression
Romano, Y., Patterson, E., & Candès, E. J. (2019) · arXiv · doi:10.48550/arxiv.1905.03222 · Cited by 236 · see citing works
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An Exact and Robust Conformal Inference Method for Counterfactual and Synthetic Controls
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Feature Importance in Gradient Boosting Trees with Cross-Validation Feature Selection
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Explainable AI for Healthcare: From Black Box to Interpretable Models
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Explainable Artificial Intelligence for Tabular Data: A Survey
Sahakyan, M., Aung, Z., & Rahwan, T. (2021) · IEEE Access · doi:10.1109/access.2021.3116481 · Cited by 135 · see citing works
“Another model-agnostic technique for outcome explanation is LOCO (Leave One Covariate Out) [32], which scores the instance features by repeatedly running the model, each time leaving one feature out.”
MethodCross-Validation With Confidence
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Angelopoulos, A. N., Bates, S., Fisch, A., Lei, L., & Schuster, T. (2022) · arXiv · doi:10.48550/arxiv.2208.02814 · Cited by 37 · see citing works
Valid Model-Free Spatial Prediction
Mao, H., Martin, R., & Reich, B. J. (2024) · Journal of the American Statistical Association · doi:10.1080/01621459.2022.2147531 · Cited by 28 · see citing works
Truck traffic monitoring with satellite images
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Testing conditional independence in supervised learning algorithms
Watson, D. S., & Wright, M. N. (2021) · Machine Learning · doi:10.1007/s10994-021-06030-6 · Cited by 13 · see citing works
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ConfIC-RCA: Statistically Grounded Efficient Estimation of Segmentation Quality
Cosarinsky, M., Billot, R., Mansilla, L., Jimenez, G., Gaggión, N., Fu, G., Tirer, T., & Ferrante, E. (2025) · arXiv
“To better account for heteroscedasticity, i.e. when the noise variance varies across the input domain, a locally weighted nonconformity score has been proposed: s ( x , y ) = | y − ˆ f ( x ) | ˆ σ ( x ) , where ˆ σ ( x ) is the estimated standard error of the residual [43].”
- Authors
- Lei, J., G’Sell, M., Rinaldo, A., Tibshirani, R. J., & Wasserman, L.
- Affiliations
- Carnegie Mellon University (US)
- Venue
- Journal of the American Statistical Association
- Publisher
- Informa UK Limited
- Published
- 2018-06-08
- Volume
- 113
- Issue
- 523
- Pages
- 1094-1111
- ISSN
- 0162-1459 (ISSN-L)
- 1537-274X
- OpenAlex
- W2964060211
- Cited by
- 1,049
- References
- 3
- Citations by year
- 2026: 417 · 2025: 168 · 2024: 111 · 2023: 94 · 2022: 73 · 2021: 83 · 2020: 58 · 2019: 31 · 2018: 7 · 2017: 4 · 2016: 2 · 2012: 1
- FWCI
- 18.32
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Journal of the American Statistical Association (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- RePEc: Research Papers in Economics (submittedVersion)
- Figshare (submittedVersion, cc-by)
- arXiv (Cornell University)
- Figshare (cc-by)
- Funders
- Division of Mathematical Sciences (1553884)
- Topics
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Inference
- Physical Sciences › Mathematics › Statistics and Probability › Advanced Statistical Methods and Models
- Physical Sciences › Mathematics › Statistics and Probability › Probability and Statistical Research
- Keywords
- conformal inference, jackknife, prediction bands, prediction intervals, heteroscedasticity, feature importance, model-free inference
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