# Integral Imprecise Probability Metrics

arXiv: https://arxiv.org/abs/2505.16156 · HTML: https://mlsanity.com/paper/2505.16156/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 48 of its 156 references that are in the map.
- First posted on arXiv 22 May 2025.

## Builds on (most cited first, 30 of 48)

- [What Uncertainties Do We Need in Bayesian Deep Learning for Computer Vision?](https://mlsanity.com/paper/1703.04977/) (2017)
- [A Survey on Hallucination in Large Language Models: Principles, Taxonomy, Challenges, and Open Questions](https://mlsanity.com/paper/2311.05232/) (2023)
- [Aleatoric and Epistemic Uncertainty in Machine Learning: An Introduction to Concepts and Methods](https://mlsanity.com/paper/1910.09457/) (2019)
- [Evidential Deep Learning to Quantify Classification Uncertainty](https://mlsanity.com/paper/1806.01768/) (2018)
- [Domain Generalization via Invariant Feature Representation](https://mlsanity.com/paper/1301.2115/) (2013)
- [Kernel Mean Embedding of Distributions: A Review and Beyond](https://mlsanity.com/paper/1605.09522/) (2016)
- [MMD GAN: Towards Deeper Understanding of Moment Matching Network](https://mlsanity.com/paper/1705.08584/) (2017)
- [Maximum Mean Discrepancy Gradient Flow](https://mlsanity.com/paper/1906.04370/) (2019)
- [Quantifying Aleatoric and Epistemic Uncertainty in Machine Learning: Are Conditional Entropy and Mutual Information Appropriate Measures?](https://mlsanity.com/paper/2209.03302/) (2022)
- [Learning Theory for Distribution Regression](https://mlsanity.com/paper/1411.2066/) (2014)
- [Statistical Inference for Generative Models with Maximum Mean Discrepancy](https://mlsanity.com/paper/1906.05944/) (2019)
- [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)
- [Rethinking Aleatoric and Epistemic Uncertainty](https://mlsanity.com/paper/2412.20892/) (2024)
- [Is the Volume of a Credal Set a Good Measure for Epistemic Uncertainty?](https://mlsanity.com/paper/2306.09586/) (2023)
- [Metrizing Weak Convergence with Maximum Mean Discrepancies](https://mlsanity.com/paper/2006.09268/) (2020)
- [MMD-Bayes: Robust Bayesian Estimation via Maximum Mean Discrepancy](https://mlsanity.com/paper/1909.13339/) (2019)
- [Second-Order Uncertainty Quantification: A Distance-Based Approach](https://mlsanity.com/paper/2312.00995/) (2023)
- [ROAD-R: The Autonomous Driving Dataset with Logical Requirements](https://mlsanity.com/paper/2210.01597/) (2022)
- [Counterfactual Mean Embeddings](https://mlsanity.com/paper/1805.08845/) (2018)
- [Conformal prediction under ambiguous ground truth](https://mlsanity.com/paper/2307.09302/) (2023)
- [RKHS-SHAP: Shapley Values for Kernel Methods](https://mlsanity.com/paper/2110.09167/) (2021)
- [CreINNs: Credal-Set Interval Neural Networks for Uncertainty Estimation in Classification Tasks](https://mlsanity.com/paper/2401.05043/) (2024)
- [Deconditional Downscaling with Gaussian Processes](https://mlsanity.com/paper/2105.12909/) (2021)
- [BayesIMP: Uncertainty Quantification for Causal Data Fusion](https://mlsanity.com/paper/2106.03477/) (2021)
- [Explaining the Uncertain: Stochastic Shapley Values for Gaussian Process Models](https://mlsanity.com/paper/2305.15167/) (2023)
- [Random-Set Neural Networks (RS-NN)](https://mlsanity.com/paper/2307.05772/) (2023)
- [Second-Order Uncertainty Quantification: Variance-Based Measures](https://mlsanity.com/paper/2401.00276/) (2023)
- [Conformalized Credal Regions for Classification with Ambiguous Ground Truth](https://mlsanity.com/paper/2411.04852/) (2024)
- [Ensemble-based Uncertainty Quantification: Bayesian versus Credal Inference](https://mlsanity.com/paper/2107.10384/) (2021)

## Built on by (most cited first, 11 of 11)

- [Credal and Interval Deep Evidential Classifications](https://mlsanity.com/paper/2512.05526/) (2025)
- [Learning Credal Ensembles via Distributionally Robust Optimization](https://mlsanity.com/paper/2602.08470/) (2026)
- [Verbalizing LLM's Higher-order Uncertainty via Imprecise Probabilities](https://mlsanity.com/paper/2603.10396/) (2026)
- [Beyond Prediction-Set Size: Quantifying Conformal Uncertainty through Credal Geometry](https://mlsanity.com/paper/2602.01667/) (2026)
- [Quantification of Credal Uncertainty: A Distance-Based Approach](https://mlsanity.com/paper/2603.27270/) (2026)
- [When Do Credal Sets Stabilize? Fixed-Point Theorems for Credal Set Updates](https://mlsanity.com/paper/2510.04769/) (2025)
- [Maxitive Donsker-Varadhan Formulation for Possibilistic Variational Inference](https://mlsanity.com/paper/2511.21223/) (2025)
- [Set-based v.s. Distribution-based Representations of Epistemic Uncertainty: A Comparative Study](https://mlsanity.com/paper/2602.22747/) (2026)
- [Efficient Credal Prediction through Decalibration](https://mlsanity.com/paper/2603.08495/) (2026)
- [Upper Entropy for 2-Monotone Lower Probabilities](https://mlsanity.com/paper/2603.23558/) (2026)
- [Uncertainty quantification for trustworthy deep learning: Methods and measures](https://mlsanity.com/paper/2607.28248/) (2026)

## Abstract (arXiv)

Quantifying differences between probability distributions is fundamental to statistics and machine learning, primarily for comparing statistical uncertainty. In contrast, epistemic uncertainty -- due to incomplete knowledge -- requires richer representations than those offered by classical probability. Imprecise probability (IP) theory offers such models, capturing ambiguity and partial belief. This has driven growing interest in imprecise probabilistic machine learning (IPML), where inference and decision-making rely on broader uncertainty models -- highlighting the need for metrics beyond classical probability. This work introduces the integral imprecise probability metric framework, a Choquet integral-based generalisation of classical integral probability metrics to the setting of capacities -- a broad class of IP models encompassing many existing ones, including lower probabilities, probability intervals, belief functions, and more. Theoretically, we establish conditions under which IIPM serves as a valid metric and metrises a form of weak convergence of capacities. Practically, IIPM not only enables comparison across different IP models but also supports the quantification of epistemic uncertainty~(EU) within a single IP model. In particular, by comparing an IP model with its conjugate, IIPM gives rise to a new class of epistemic uncertainty measures -- Maximum Mean Imprecision -- which satisfy key axiomatic properties proposed in the uncertainty quantification literature. We validate MMI through selective classification experiments, demonstrating strong empirical performance against established EU measures, and outperforming them when classical methods struggle to scale to a large number of classes. Our work advances both theory and practice in Imprecise Probabilistic Machine Learning, offering a principled framework for comparing and quantifying epistemic uncertainty under imprecision.