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Bengs, Viktor ORCID logoORCID: https://orcid.org/0000-0001-6988-6186; Hüllermeier, Eyke ORCID logoORCID: https://orcid.org/0000-0002-9944-4108 and Waegeman, Willem ORCID logoORCID: https://orcid.org/0000-0002-5950-3003 (2023): On Second-Order Scoring Rules for Epistemic Uncertainty Quantification. 40th International Conference on Machine Learning (ICML 2023), Hawaii, USA, 23-29 July, 2023. Krause, Andreas; Brunskill, Emma; Cho, Kyunghyun; Engelhardt, Barbara; Sabato, Sivan and Scarlett, Jonathan (eds.) : In: Proceedings of the 40th International Conference on Machine Learning, Vol. 202 PMLR. pp. 2078-2091

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Abstract

It is well known that accurate probabilistic predictors can be trained through empirical risk minimisation with proper scoring rules as loss functions. While such learners capture so-called aleatoric uncertainty of predictions, various machine learning methods have recently been developed with the goal to let the learner also represent its epistemic uncertainty, i.e., the uncertainty caused by a lack of knowledge and data. An emerging branch of the literature proposes the use of a second-order learner that provides predictions in terms of distributions on probability distributions. However, recent work has revealed serious theoretical shortcomings for second-order predictors based on loss minimisation. In this paper, we generalise these findings and prove a more fundamental result: There seems to be no loss function that provides an incentive for a second-order learner to faithfully represent its epistemic uncertainty in the same manner as proper scoring rules do for standard (first-order) learners. As a main mathematical tool to prove this result, we introduce the generalised notion of second-order scoring rules.

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