# From Risk to Uncertainty: Generating Predictive Uncertainty Measures via Bayesian Estimation

arXiv: https://arxiv.org/abs/2402.10727 · HTML: https://mlsanity.com/paper/2402.10727/

## Key facts

- Cited by 28 papers in the map (data through 14 Oct 2026).
- Builds on 38 of its 55 references that are in the map.
- First posted on arXiv 16 Feb 2024.

## Builds on (most cited first, 30 of 38)

- [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)
- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [What Uncertainties Do We Need in Bayesian Deep Learning for Computer Vision?](https://mlsanity.com/paper/1703.04977/) (2017)
- [The Many Faces of Robustness: A Critical Analysis of Out-of-Distribution Generalization](https://mlsanity.com/paper/2006.16241/) (2020)
- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Natural Adversarial Examples](https://mlsanity.com/paper/1907.07174/) (2019)
- [Energy-based Out-of-distribution Detection](https://mlsanity.com/paper/2010.03759/) (2020)
- [A Survey on Deep Learning in Medical Image Analysis](https://mlsanity.com/paper/1702.05747/) (2017)
- [Aleatoric and Epistemic Uncertainty in Machine Learning: An Introduction to Concepts and Methods](https://mlsanity.com/paper/1910.09457/) (2019)
- [Selective Classification for Deep Neural Networks](https://mlsanity.com/paper/1705.08500/) (2017)
- [Deep Bayesian Active Learning with Image Data](https://mlsanity.com/paper/1703.02910/) (2017)
- [Predictive Uncertainty Estimation via Prior Networks](https://mlsanity.com/paper/1802.10501/) (2018)
- [Your Classifier is Secretly an Energy Based Model and You Should Treat it Like One](https://mlsanity.com/paper/1912.03263/) (2019)
- [A Survey of Deep Learning Techniques for Autonomous Driving](https://mlsanity.com/paper/1910.07738/) (2019)
- [Deep Learning for Medical Image Analysis](https://mlsanity.com/paper/1708.08987/) (2017)
- [Simple and Principled Uncertainty Estimation with Deterministic Deep Learning via Distance Awareness](https://mlsanity.com/paper/2006.10108/) (2020)
- [What Are Bayesian Neural Network Posteriors Really Like?](https://mlsanity.com/paper/2104.14421/) (2021)
- [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)
- [Deep Learning for Financial Applications : A Survey](https://mlsanity.com/paper/2002.05786/) (2020)
- [Estimating Epistemic and Aleatoric Uncertainty with a Single Model](https://mlsanity.com/paper/2402.03478/) (2024)
- [On Uncertainty, Tempering, and Data Augmentation in Bayesian Classification](https://mlsanity.com/paper/2203.16481/) (2022)
- [Accurate Uncertainty Estimation and Decomposition in Ensemble Learning](https://mlsanity.com/paper/1911.04061/) (2019)
- [On Second-Order Scoring Rules for Epistemic Uncertainty Quantification](https://mlsanity.com/paper/2301.12736/) (2023)
- [Monte Carlo Variational Auto-Encoders](https://mlsanity.com/paper/2106.15921/) (2021)
- [Nonparametric Uncertainty Quantification for Single Deterministic Neural Network](https://mlsanity.com/paper/2202.03101/) (2022)
- [FedPop: A Bayesian Approach for Personalised Federated Learning](https://mlsanity.com/paper/2206.03611/) (2022)
- [Quantification of Uncertainty with Adversarial Models](https://mlsanity.com/paper/2307.03217/) (2023)

## Built on by (most cited first, 28 of 28)

- [Rethinking Aleatoric and Epistemic Uncertainty](https://mlsanity.com/paper/2412.20892/) (2024)
- [Uncertainty Quantification for LLMs through Minimum Bayes Risk: Bridging Confidence and Consistency](https://mlsanity.com/paper/2502.04964/) (2025)
- [Quantifying Aleatoric and Epistemic Uncertainty with Proper Scoring Rules](https://mlsanity.com/paper/2404.12215/) (2024)
- [Rethinking Uncertainty Estimation in LLMs: A Principled Single-Sequence Measure](https://mlsanity.com/paper/2412.15176/) (2024)
- [The Illusion of Certainty: Uncertainty Quantification for LLMs Fails under Ambiguity](https://mlsanity.com/paper/2511.04418/) (2025)
- [On Information-Theoretic Measures of Predictive Uncertainty](https://mlsanity.com/paper/2410.10786/) (2024)
- [Uncertainty Quantification for Hallucination Detection in Large Language Models: Foundations, Methodology, and Future Directions](https://mlsanity.com/paper/2510.12040/) (2025)
- [Provable Uncertainty Decomposition via Higher-Order Calibration](https://mlsanity.com/paper/2412.18808/) (2024)
- [An Axiomatic Assessment of Entropy- and Variance-based Uncertainty Quantification in Regression](https://mlsanity.com/paper/2504.18433/) (2025)
- [Addressing Pitfalls in the Evaluation of Uncertainty Estimation Methods for Natural Language Generation](https://mlsanity.com/paper/2510.02279/) (2025)
- [Uncertainty as Feature Gaps: Epistemic Uncertainty Quantification of LLMs in Contextual Question-Answering](https://mlsanity.com/paper/2510.02671/) (2025)
- [ReDAct: Uncertainty-Aware Deferral for LLM Agents](https://mlsanity.com/paper/2604.07036/) (2026)
- [UNCERTAINTY-LINE: Length-Invariant Estimation of Uncertainty for Large Language Models](https://mlsanity.com/paper/2505.19060/) (2025)
- [Uncertainty Quantification for Regression using Proper Scoring Rules](https://mlsanity.com/paper/2509.26610/) (2025)
- [Verbalizing LLM's Higher-order Uncertainty via Imprecise Probabilities](https://mlsanity.com/paper/2603.10396/) (2026)
- [Uncertainty Quantification for Machine Learning: One Size Does Not Fit All](https://mlsanity.com/paper/2512.12341/) (2025)
- [Deep Ensembles for Epistemic Uncertainty: A Frequentist Perspective](https://mlsanity.com/paper/2510.22063/) (2025)
- [Uncertainty Quantification for Regression: A Unified Framework based on kernel scores](https://mlsanity.com/paper/2510.25599/) (2025)
- [Evaluating Epistemic Uncertainty: Beyond OOD Detection and Active Learning](https://mlsanity.com/paper/2607.14817/) (2026)
- [Uncertainty Quantification with Proper Scoring Rules: Adjusting Measures to Prediction Tasks](https://mlsanity.com/paper/2505.22538/) (2025)
- [Multidimensional Uncertainty Quantification via Optimal Transport](https://mlsanity.com/paper/2509.22380/) (2025)
- [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)
- [Code-MUE: Measuring Code LLMs' Uncertainty through Execution-based Semantic Interaction Graphs](https://mlsanity.com/paper/2607.12273/) (2026)
- [Decomposing LLM-Judge Uncertainty to Target Expert Labels](https://mlsanity.com/paper/2609.06444/) (2026)
- [Uncertainty Quantification for LLM Agents: A Taxonomy, an Evaluation Protocol, and an Empirical Study](https://mlsanity.com/paper/2609.07395/) (2026)
- [Uncertainty Evaluation: Bridging Machine Learning and Measurement Science](https://mlsanity.com/paper/2610.18066/) (2026)

## Abstract (arXiv)

There are various measures of predictive uncertainty in the literature, but their relationships to each other remain unclear. This paper uses a decomposition of statistical pointwise risk into components, associated with different sources of predictive uncertainty, namely aleatoric uncertainty (inherent data variability) and epistemic uncertainty (model-related uncertainty). Together with Bayesian methods, applied as an approximation, we build a framework that allows one to generate different predictive uncertainty measures. We validate our method on image datasets by evaluating its performance in detecting out-of-distribution and misclassified instances using the AUROC metric. The experimental results confirm that the measures derived from our framework are useful for the considered downstream tasks.