An Axiomatic Assessment of Entropy- and Variance-based Uncertainty Quantification in Regression

Christopher Bülte, Yusuf Sale, Timo Löhr, Paul Hofman, Gitta Kutyniok, Eyke Hüllermeier
Proceedings of the 42nd Conference on Uncertainty in Artificial Intelligence, PMLR 337:880-899, 2026.

Abstract

Uncertainty quantification is crucial in machine learning, yet most (axiomatic) studies of uncertainty measures focus on classification, leaving a gap in regression settings with limited formal justification and evaluations. In this work, we provide a formal way of representing uncertainty in continuous space, using a general parametric formulation, allowing for tractable analysis and evaluation of uncertainty measures. Within this framework, we propose a set of axioms that enable rigorous assessment of total, aleatoric, and epistemic uncertainty measures. Together, this allows for a theoretical examination of uncertainty measures and their corresponding properties. As a specific example, we compare the widely used entropy- and variance-based measures with respect to established predictive models and analyze their limitations and challenges in uncertainty quantification. Our work provides a principled way to understand and develop uncertainty measures in supervised regression, offering theoretical insights and practical guidelines for reliable uncertainty assessment.

Cite this Paper

BibTeX
@InProceedings{pmlr-v337-bulte26b, title = {An Axiomatic Assessment of Entropy- and Variance-based Uncertainty Quantification in Regression}, author = {B\"{u}lte, Christopher and Sale, Yusuf and L\"{o}hr, Timo and Hofman, Paul and Kutyniok, Gitta and H\"{u}llermeier, Eyke}, booktitle = {Proceedings of the 42nd Conference on Uncertainty in Artificial Intelligence}, pages = {880--899}, year = {2026}, editor = {Perković, Emilija and Malinsky, Daniel}, volume = {337}, series = {Proceedings of Machine Learning Research}, month = {17--21 Aug}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v337/main/assets/bulte26b/bulte26b.pdf}, url = {https://proceedings.mlr.press/v337/bulte26b.html}, abstract = {Uncertainty quantification is crucial in machine learning, yet most (axiomatic) studies of uncertainty measures focus on classification, leaving a gap in regression settings with limited formal justification and evaluations. In this work, we provide a formal way of representing uncertainty in continuous space, using a general parametric formulation, allowing for tractable analysis and evaluation of uncertainty measures. Within this framework, we propose a set of axioms that enable rigorous assessment of total, aleatoric, and epistemic uncertainty measures. Together, this allows for a theoretical examination of uncertainty measures and their corresponding properties. As a specific example, we compare the widely used entropy- and variance-based measures with respect to established predictive models and analyze their limitations and challenges in uncertainty quantification. Our work provides a principled way to understand and develop uncertainty measures in supervised regression, offering theoretical insights and practical guidelines for reliable uncertainty assessment.} }
Endnote
%0 Conference Paper %T An Axiomatic Assessment of Entropy- and Variance-based Uncertainty Quantification in Regression %A Christopher Bülte %A Yusuf Sale %A Timo Löhr %A Paul Hofman %A Gitta Kutyniok %A Eyke Hüllermeier %B Proceedings of the 42nd Conference on Uncertainty in Artificial Intelligence %C Proceedings of Machine Learning Research %D 2026 %E Emilija Perković %E Daniel Malinsky %F pmlr-v337-bulte26b %I PMLR %P 880--899 %U https://proceedings.mlr.press/v337/bulte26b.html %V 337 %X Uncertainty quantification is crucial in machine learning, yet most (axiomatic) studies of uncertainty measures focus on classification, leaving a gap in regression settings with limited formal justification and evaluations. In this work, we provide a formal way of representing uncertainty in continuous space, using a general parametric formulation, allowing for tractable analysis and evaluation of uncertainty measures. Within this framework, we propose a set of axioms that enable rigorous assessment of total, aleatoric, and epistemic uncertainty measures. Together, this allows for a theoretical examination of uncertainty measures and their corresponding properties. As a specific example, we compare the widely used entropy- and variance-based measures with respect to established predictive models and analyze their limitations and challenges in uncertainty quantification. Our work provides a principled way to understand and develop uncertainty measures in supervised regression, offering theoretical insights and practical guidelines for reliable uncertainty assessment.
APA
Bülte, C., Sale, Y., Löhr, T., Hofman, P., Kutyniok, G. & Hüllermeier, E.. (2026). An Axiomatic Assessment of Entropy- and Variance-based Uncertainty Quantification in Regression. Proceedings of the 42nd Conference on Uncertainty in Artificial Intelligence, in Proceedings of Machine Learning Research 337:880-899 Available from https://proceedings.mlr.press/v337/bulte26b.html.

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