# Calibration Prediction Interval for Non-parametric Regression and Neural Networks

arXiv: https://arxiv.org/abs/2509.02735 · HTML: https://mlsanity.com/paper/2509.02735/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 39 references that are in the map.
- First posted on arXiv 2 Sep 2025.

## Builds on (most cited first, 15 of 15)

- [On Calibration of Modern Neural Networks](https://mlsanity.com/paper/1706.04599/) (2017)
- [Distribution-Free Predictive Inference For Regression](https://mlsanity.com/paper/1604.04173/) (2016)
- [Nonparametric regression using deep neural networks with ReLU activation function](https://mlsanity.com/paper/1708.06633/) (2017)
- [Nearly-tight VC-dimension and pseudodimension bounds for piecewise linear neural networks](https://mlsanity.com/paper/1703.02930/) (2017)
- [Deep Network Approximation Characterized by Number of Neurons](https://mlsanity.com/paper/1906.05497/) (2019)
- [High-Quality Prediction Intervals for Deep Learning: A Distribution-Free, Ensembled Approach](https://mlsanity.com/paper/1802.07167/) (2018)
- [Deep ReLU network approximation of functions on a manifold](https://mlsanity.com/paper/1908.00695/) (2019)
- [Wasserstein Generative Learning of Conditional Distribution](https://mlsanity.com/paper/2112.10039/) (2021)
- [Methods to Compute Prediction Intervals: A Review and New Results](https://mlsanity.com/paper/2011.03065/) (2020)
- [PIVEN: A Deep Neural Network for Prediction Intervals with Specific Value Prediction](https://mlsanity.com/paper/2006.05139/) (2020)
- [Scalable Subsampling Inference for Deep Neural Networks](https://mlsanity.com/paper/2405.08276/) (2024)
- [Model-free Bootstrap and Conformal Prediction in Regression: Conditionality, Conjecture Testing, and Pertinent Prediction Intervals](https://mlsanity.com/paper/2109.12156/) (2021)
- [Prediction intervals for Deep Neural Networks](https://mlsanity.com/paper/2010.04044/) (2020)
- [Confidence Interval Construction and Conditional Variance Estimation with Dense ReLU Networks](https://mlsanity.com/paper/2412.20355/) (2024)
- [Deep Limit Model-free Prediction in Regression](https://mlsanity.com/paper/2408.09532/) (2024)

## Abstract (arXiv)

Accurate conditional prediction in the regression setting plays an important role in many real-world problems. Typically, a point prediction often falls short since no attempt is made to quantify the prediction accuracy. Classically, under the normality and linearity assumptions, the Prediction Interval (PI) for the response variable can be determined routinely based on the t distribution. Unfortunately, these two assumptions are rarely met in practice. To fully avoid these two conditions, we develop a so-called calibration PI (cPI) which leverages estimations by Deep Neural Networks (DNN) or kernel methods. Moreover, the cPI can be easily adjusted to capture the estimation variability within the prediction procedure, which is a crucial error source often ignored in practice. Under regular assumptions, we verify that our cPI has an asymptotically valid coverage rate. We also demonstrate that cPI based on the kernel method ensures a coverage rate with a high probability when the sample size is large. Besides, with several conditions, the cPI based on DNN works even with finite samples. A comprehensive simulation study supports the usefulness of cPI, and the convincing performance of cPI with a short sample is confirmed with two empirical datasets.