# Is the Volume of a Credal Set a Good Measure for Epistemic Uncertainty?

arXiv: https://arxiv.org/abs/2306.09586 · HTML: https://mlsanity.com/paper/2306.09586/

## Key facts

- Cited by 40 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 53 references that are in the map.
- First posted on arXiv 16 Jun 2023.

## Builds on (most cited first, 10 of 10)

- [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)
- [Understanding Measures of Uncertainty for Adversarial Example Detection](https://mlsanity.com/paper/1803.08533/) (2018)
- [Quantifying Aleatoric and Epistemic Uncertainty in Machine Learning: Are Conditional Entropy and Mutual Information Appropriate Measures?](https://mlsanity.com/paper/2209.03302/) (2022)
- [Pitfalls of Epistemic Uncertainty Quantification through Loss Minimisation](https://mlsanity.com/paper/2203.06102/) (2022)
- [On the Safety of Machine Learning: Cyber-Physical Systems, Decision Sciences, and Data Products](https://mlsanity.com/paper/1610.01256/) (2016)
- [On Uncertainty, Tempering, and Data Augmentation in Bayesian Classification](https://mlsanity.com/paper/2203.16481/) (2022)
- [Credal Bayesian Deep Learning](https://mlsanity.com/paper/2302.09656/) (2023)
- [Aleatoric and Epistemic Uncertainty with Random Forests](https://mlsanity.com/paper/2001.00893/) (2020)
- [Engineering Safety in Machine Learning](https://mlsanity.com/paper/1601.04126/) (2016)

## Built on by (most cited first, 30 of 40)

- [Rethinking Aleatoric and Epistemic Uncertainty](https://mlsanity.com/paper/2412.20892/) (2024)
- [Second-Order Uncertainty Quantification: A Distance-Based Approach](https://mlsanity.com/paper/2312.00995/) (2023)
- [Credal Learning Theory](https://mlsanity.com/paper/2402.00957/) (2024)
- [Introducing an Improved Information-Theoretic Measure of Predictive Uncertainty](https://mlsanity.com/paper/2311.08309/) (2023)
- [CreINNs: Credal-Set Interval Neural Networks for Uncertainty Estimation in Classification Tasks](https://mlsanity.com/paper/2401.05043/) (2024)
- [Label-wise Aleatoric and Epistemic Uncertainty Quantification](https://mlsanity.com/paper/2406.02354/) (2024)
- [Conformalized Credal Set Predictors](https://mlsanity.com/paper/2402.10723/) (2024)
- [Random-Set Neural Networks (RS-NN)](https://mlsanity.com/paper/2307.05772/) (2023)
- [Conformalized Credal Regions for Classification with Ambiguous Ground Truth](https://mlsanity.com/paper/2411.04852/) (2024)
- [On Information-Theoretic Measures of Predictive Uncertainty](https://mlsanity.com/paper/2410.10786/) (2024)
- [A Novel Bayes' Theorem for Upper Probabilities](https://mlsanity.com/paper/2307.06831/) (2023)
- [Credal Wrapper of Model Averaging for Uncertainty Estimation in Classification](https://mlsanity.com/paper/2405.15047/) (2024)
- [Integral Imprecise Probability Metrics](https://mlsanity.com/paper/2505.16156/) (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)
- [Epistemic Artificial Intelligence is Essential for Machine Learning Models to Truly 'Know When They Do Not Know'](https://mlsanity.com/paper/2505.04950/) (2025)
- [Credal and Interval Deep Evidential Classifications](https://mlsanity.com/paper/2512.05526/) (2025)
- [Conformal Prediction Regions are Imprecise Highest Density Regions](https://mlsanity.com/paper/2502.06331/) (2025)
- [A calibration test for evaluating set-based epistemic uncertainty representations](https://mlsanity.com/paper/2502.16299/) (2025)
- [Random-Set Large Language Models](https://mlsanity.com/paper/2504.18085/) (2025)
- [Learning Credal Ensembles via Distributionally Robust Optimization](https://mlsanity.com/paper/2602.08470/) (2026)
- [Truthful Elicitation of Imprecise Forecasts](https://mlsanity.com/paper/2503.16395/) (2025)
- [Generalising realisability in statistical learning theory under epistemic uncertainty](https://mlsanity.com/paper/2402.14759/) (2024)
- [Epistemic Wrapping for Uncertainty Quantification](https://mlsanity.com/paper/2505.02277/) (2025)
- [Beyond Prediction-Set Size: Quantifying Conformal Uncertainty through Credal Geometry](https://mlsanity.com/paper/2602.01667/) (2026)
- [Partial-Label Learning with a Reject Option](https://mlsanity.com/paper/2402.00592/) (2024)
- [Random-Set Graph Neural Networks](https://mlsanity.com/paper/2605.11987/) (2026)
- [Towards Better Understanding of In-Context Learning Ability from In-Context Uncertainty Quantification](https://mlsanity.com/paper/2405.15115/) (2024)
- [Generalized Regularized Evidential Deep Learning Models: Theory and Comprehensive Evaluation](https://mlsanity.com/paper/2512.23753/) (2025)
- [Quantification of Credal Uncertainty: A Distance-Based Approach](https://mlsanity.com/paper/2603.27270/) (2026)

## Abstract (arXiv)

Adequate uncertainty representation and quantification have become imperative in various scientific disciplines, especially in machine learning and artificial intelligence. As an alternative to representing uncertainty via one single probability measure, we consider credal sets (convex sets of probability measures). The geometric representation of credal sets as d-dimensional polytopes implies a geometric intuition about (epistemic) uncertainty. In this paper, we show that the volume of the geometric representation of a credal set is a meaningful measure of epistemic uncertainty in the case of binary classification, but less so for multi-class classification. Our theoretical findings highlight the crucial role of specifying and employing uncertainty measures in machine learning in an appropriate way, and for being aware of possible pitfalls.