# Uncertainty Quantification for Regression using Proper Scoring Rules

arXiv: https://arxiv.org/abs/2509.26610 · HTML: https://mlsanity.com/paper/2509.26610/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 47 references that are in the map.
- First posted on arXiv 30 Sep 2025.

## Builds on (most cited first, 24 of 24)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [The Cityscapes Dataset for Semantic Urban Scene Understanding](https://mlsanity.com/paper/1604.01685/) (2016)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [Gaussian Error Linear Units (GELUs)](https://mlsanity.com/paper/1606.08415/) (2016)
- [What Uncertainties Do We Need in Bayesian Deep Learning for Computer Vision?](https://mlsanity.com/paper/1703.04977/) (2017)
- [Aleatoric and Epistemic Uncertainty in Machine Learning: An Introduction to Concepts and Methods](https://mlsanity.com/paper/1910.09457/) (2019)
- [BatchBALD: Efficient and Diverse Batch Acquisition for Deep Bayesian Active Learning](https://mlsanity.com/paper/1906.08158/) (2019)
- [Deep Evidential Regression](https://mlsanity.com/paper/1910.02600/) (2019)
- [Deep Deterministic Uncertainty: A Simple Baseline](https://mlsanity.com/paper/2102.11582/) (2021)
- [Quantifying Aleatoric and Epistemic Uncertainty in Machine Learning: Are Conditional Entropy and Mutual Information Appropriate Measures?](https://mlsanity.com/paper/2209.03302/) (2022)
- [DEUP: Direct Epistemic Uncertainty Prediction](https://mlsanity.com/paper/2102.08501/) (2021)
- [A Deeper Look into Aleatoric and Epistemic Uncertainty Disentanglement](https://mlsanity.com/paper/2204.09308/) (2022)
- [Reliable training and estimation of variance networks](https://mlsanity.com/paper/1906.03260/) (2019)
- [Uncertainty Quantification of Collaborative Detection for Self-Driving](https://mlsanity.com/paper/2209.08162/) (2022)
- [Nonparametric Uncertainty Quantification for Single Deterministic Neural Network](https://mlsanity.com/paper/2202.03101/) (2022)
- [Second-Order Uncertainty Quantification: A Distance-Based Approach](https://mlsanity.com/paper/2312.00995/) (2023)
- [Is Epistemic Uncertainty Faithfully Represented by Evidential Deep Learning Methods?](https://mlsanity.com/paper/2402.09056/) (2024)
- [From Risk to Uncertainty: Generating Predictive Uncertainty Measures via Bayesian Estimation](https://mlsanity.com/paper/2402.10727/) (2024)
- [Quantifying Aleatoric and Epistemic Uncertainty with Proper Scoring Rules](https://mlsanity.com/paper/2404.12215/) (2024)
- [Proper scoring rules for estimation and forecast evaluation](https://mlsanity.com/paper/2504.01781/) (2025)
- [On Information-Theoretic Measures of Predictive Uncertainty](https://mlsanity.com/paper/2410.10786/) (2024)
- [An Axiomatic Assessment of Entropy- and Variance-based Uncertainty Quantification in Regression](https://mlsanity.com/paper/2504.18433/) (2025)

## Built on by (most cited first, 5 of 5)

- [Uncertainty Quantification for Regression: A Unified Framework based on kernel scores](https://mlsanity.com/paper/2510.25599/) (2025)
- [Reinforcement Learning over Predictive Distributions for LLM Regression](https://mlsanity.com/paper/2605.20740/) (2026)
- [Subjective Risk Decomposition: A New View for Uncertainty Quantification](https://mlsanity.com/paper/2607.15196/) (2026)
- [A Ground-Truth Framework for Uncertainty Disentanglement with Posterior Risk](https://mlsanity.com/paper/2608.05995/) (2026)
- [On the QUEST for Uncertainty Quantification via Highest Density Regions](https://mlsanity.com/paper/2606.19569/) (2026)

## Abstract (arXiv)

Quantifying uncertainty of machine learning model predictions is essential for reliable decision-making, especially in safety-critical applications. Recently, uncertainty quantification (UQ) theory has advanced significantly, building on a firm basis of learning with proper scoring rules. However, these advances were focused on classification, while extending these ideas to regression remains challenging. In this work, we introduce a unified UQ framework for regression based on proper scoring rules, such as CRPS, logarithmic, squared error, and quadratic scores. We derive closed-form expressions for the resulting uncertainty measures under practical parametric assumptions and show how to estimate them using ensembles of models. In particular, the derived uncertainty measures naturally decompose into aleatoric and epistemic components. The framework recovers popular regression UQ measures based on predictive variance and differential entropy. Our broad evaluation on synthetic and real-world regression datasets provides guidance for selecting reliable UQ measures.