On Second-Order Scoring Rules for Epistemic Uncertainty Quantification

Viktor Bengs, Eyke Hüllermeier, Willem Waegeman
Proceedings of the 40th International Conference on Machine Learning, PMLR 202:2078-2091, 2023.

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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v202-bengs23a, title = {On Second-Order Scoring Rules for Epistemic Uncertainty Quantification}, author = {Bengs, Viktor and H\"{u}llermeier, Eyke and Waegeman, Willem}, booktitle = {Proceedings of the 40th International Conference on Machine Learning}, pages = {2078--2091}, year = {2023}, editor = {Krause, Andreas and Brunskill, Emma and Cho, Kyunghyun and Engelhardt, Barbara and Sabato, Sivan and Scarlett, Jonathan}, volume = {202}, series = {Proceedings of Machine Learning Research}, month = {23--29 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v202/bengs23a/bengs23a.pdf}, url = {https://proceedings.mlr.press/v202/bengs23a.html}, 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.} }
Endnote
%0 Conference Paper %T On Second-Order Scoring Rules for Epistemic Uncertainty Quantification %A Viktor Bengs %A Eyke Hüllermeier %A Willem Waegeman %B Proceedings of the 40th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2023 %E Andreas Krause %E Emma Brunskill %E Kyunghyun Cho %E Barbara Engelhardt %E Sivan Sabato %E Jonathan Scarlett %F pmlr-v202-bengs23a %I PMLR %P 2078--2091 %U https://proceedings.mlr.press/v202/bengs23a.html %V 202 %X 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.
APA
Bengs, V., Hüllermeier, E. & Waegeman, W.. (2023). On Second-Order Scoring Rules for Epistemic Uncertainty Quantification. Proceedings of the 40th International Conference on Machine Learning, in Proceedings of Machine Learning Research 202:2078-2091 Available from https://proceedings.mlr.press/v202/bengs23a.html.

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