mlsanityData to 14 Oct 2026

Second-Order Uncertainty Quantification: A Distance-Based Approach

arXiv 2312.00995 ↗First posted 2 Dec 2023cs.LG stat.MLcited by 35

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Abstract (arXiv)

In the past couple of years, various approaches to representing and quantifying different types of predictive uncertainty in machine learning, notably in the setting of classification, have been proposed on the basis of second-order probability distributions, i.e., predictions in the form of distributions on probability distributions. A completely conclusive solution has not yet been found, however, as shown by recent criticisms of commonly used uncertainty measures associated with second-order distributions, identifying undesirable theoretical properties of these measures. In light of these criticisms, we propose a set of formal criteria that meaningful uncertainty measures for predictive uncertainty based on second-order distributions should obey. Moreover, we provide a general framework for developing uncertainty measures to account for these criteria, and offer an instantiation based on the Wasserstein distance, for which we prove that all criteria are satisfied.

Key facts. Cited by 35 papers in the map (data through 14 Oct 2026). Builds on 18 of its 47 references that are in the map. First posted on arXiv 2 Dec 2023.

Builds on

18 of 47 references.

References · 18

  1. 2017What Uncertainties Do We Need in Bayesian Deep Learning for Computer Vision?cited by 2,352
  2. 2019Aleatoric and Epistemic Uncertainty in Machine Learning: An Introduction to Concepts and Methodscited by 913
  3. 2019Deep Evidential Regressioncited by 355
  4. 2018Understanding Measures of Uncertainty for Adversarial Example Detectioncited by 177
  5. 2022Quantifying Aleatoric and Epistemic Uncertainty in Machine Learning: Are Conditional Entropy and Mutual Information Appropriate Measures?cited by 125
  6. 2019DropConnect Is Effective in Modeling Uncertainty of Bayesian Deep Networkscited by 92
  7. 2021Prior and Posterior Networks: A Survey on Evidential Deep Learning Methods For Uncertainty Estimationcited by 71
  8. 2022Pitfalls of Epistemic Uncertainty Quantification through Loss Minimisationcited by 66
  9. 2016On the Safety of Machine Learning: Cyber-Physical Systems, Decision Sciences, and Data Productscited by 52
  10. 2022On Uncertainty, Tempering, and Data Augmentation in Bayesian Classificationcited by 46
  11. 2022The Unreasonable Effectiveness of Deep Evidential Regressioncited by 40
  12. 2023Is the Volume of a Credal Set a Good Measure for Epistemic Uncertainty?cited by 40
6 more
  1. 2023On Second-Order Scoring Rules for Epistemic Uncertainty Quantificationcited by 38
  2. 2020Regression Prior Networkscited by 28
  3. 2023Introducing an Improved Information-Theoretic Measure of Predictive Uncertaintycited by 28
  4. 2016Engineering Safety in Machine Learningcited by 24
  5. 2021Multivariate Deep Evidential Regressioncited by 23
  6. 2023A Novel Bayes' Theorem for Upper Probabilitiescited by 13

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2023: 1; 2024: 15; 2025: 12; 2026: 7.

35 papers.

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  1. 2024Rethinking Aleatoric and Epistemic Uncertaintycited by 42
  2. 2024Is Epistemic Uncertainty Faithfully Represented by Evidential Deep Learning Methods?cited by 35
  3. 2024Credal Learning Theorycited by 31
  4. 2024A Comprehensive Survey on Evidential Deep Learning and Its Applicationscited by 25
  5. 2024Label-wise Aleatoric and Epistemic Uncertainty Quantificationcited by 22
  6. 2023Second-Order Uncertainty Quantification: Variance-Based Measurescited by 19
  7. 2024Conformalized Credal Regions for Classification with Ambiguous Ground Truthcited by 19
  8. 2024On Information-Theoretic Measures of Predictive Uncertaintycited by 16
  9. 2024Revisiting Essential and Nonessential Settings of Evidential Deep Learningcited by 15
  10. 2024Scalable Bayesian Learning with posteriorscited by 13
  11. 2025Integral Imprecise Probability Metricscited by 11
  12. 2024Provable Uncertainty Decomposition via Higher-Order Calibrationcited by 9
23 more
  1. 2025An Axiomatic Assessment of Entropy- and Variance-based Uncertainty Quantification in Regressioncited by 9
  2. 2025Position: Epistemic uncertainty estimation methods are fundamentally incompletecited by 9
  3. 2025Conformal Prediction Regions are Imprecise Highest Density Regionscited by 7
  4. 2025Prior2Former -- Evidential Modeling of Mask Transformers for Assumption-Free Open-World Panoptic Segmentationcited by 7
  5. 2024Generalising realisability in statistical learning theory under epistemic uncertaintycited by 5
  6. 2024Enhancing Uncertainty Quantification in Drug Discovery with Censored Regression Labelscited by 5
  7. 2025Confidence Elicitation: A New Attack Vector for Large Language Modelscited by 5
  8. 2025Uncertainty Quantification for Regression using Proper Scoring Rulescited by 5
  9. 2024Improving Active Learning with a Bayesian Representation of Epistemic Uncertaintycited by 4
  10. 2026Beyond Prediction-Set Size: Quantifying Conformal Uncertainty through Credal Geometrycited by 4
  11. 2026Task-Awareness Improves LLM Generations and Uncertaintycited by 3
  12. 2025Uncertainty Quantification for Regression: A Unified Framework based on kernel scorescited by 2
  13. 2025Generalized Regularized Evidential Deep Learning Models: Theory and Comprehensive Evaluationcited by 2
  14. 2026Quantification of Credal Uncertainty: A Distance-Based Approachcited by 2
  15. 2024Conformal Predictions for Probabilistically Robust Scalable Machine Learning Classificationcited by 1
  16. 2025Uncertainty Quantification with Proper Scoring Rules: Adjusting Measures to Prediction Taskscited by 1
  17. 2025Epistemic Errors of Imperfect Multitask Learners When Distributions Shiftcited by 1
  18. 2026Set-based v.s. Distribution-based Representations of Epistemic Uncertainty: A Comparative Studycited by 1
  19. 2024Optimal Transport for ε-Contaminated Credal Sets: To the Memory of Sayan Mukherjeecited by 0
  20. 2025A Machine Learning Perspective on Automated Driving Corner Casescited by 0
  21. 2026The Minimax Rate of Perturbed Second-Order Calibrationcited by 0
  22. 2026On the QUEST for Uncertainty Quantification via Highest Density Regionscited by 0
  23. 2026Risk and Anomaly Identification for Distribution Network Optimal Operation Based on Reinforcement Learning and Uncertainty Quantificationcited by 0