On Second-Order Scoring Rules for Epistemic Uncertainty Quantification

• URL: http://arxiv.org/abs/2301.12736v1
• Date: Mon, 30 Jan 2023 08:59:45 GMT
• Title: On Second-Order Scoring Rules for Epistemic Uncertainty Quantification
• Authors: Viktor Bengs and Eyke H\"ullermeier and Willem Waegeman
• Abstract summary: We show that there seems to be no loss function that provides an incentive for a second-order learner to faithfully represent its uncertainty. As a main mathematical tool to prove this result, we introduce the generalised notion of second-order scoring rules.
• Score: 8.298716599039501
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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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