Pith. sign in

REVIEW 3 major objections 4 minor 1 cited by

A unified framework builds regression uncertainty measures from kernel scores, making the kernel's properties the design controls for robustness, spread-sensitivity, and scale behavior.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-04 07:25 UTC pith:NW2WFGOI

load-bearing objection Kernel scores offer a genuinely useful unifying recipe for regression uncertainty measures, but two proposition-level overclaims need tightening before this is publishable as stated. the 3 major comments →

arxiv 2510.25599 v2 pith:NW2WFGOI submitted 2025-10-29 cs.LG

Uncertainty Quantification for Regression: A Unified Framework based on kernel scores

Christopher B\"ulte , Yusuf Sale , Gitta Kutyniok , Eyke H\"ullermeier This is my paper
classification cs.LG
keywords uncertainty quantificationkernel scoresproper scoring rulesregressionaleatoric uncertaintyepistemic uncertaintyenergy scoreconvex order
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper aims to establish that total, aleatoric, and epistemic uncertainty in multivariate regression can be defined and analyzed through a single family of measures built from strictly proper kernel scores. The central claim is that the kernel is not a free parameter but a design axis: a bounded kernel yields outlier-robust uncertainty estimates, a translation-invariant convex kernel makes aleatoric uncertainty grow with the spread of the predictive distribution, and homogeneity singles out the energy score as the unique translation-invariant scale-covariant choice. The framework reproduces standard variance-based, entropy-based, and CRPS-type measures as special instances while allowing principled construction of new ones. A sympathetic reader would care because regression uncertainty quantification currently lacks such a unifying handle, and these guarantees would let practitioners pick a measure by the behavior they need.

Core claim

The paper claims that for any strictly proper scoring rule, the BMA and pairwise estimators satisfy total uncertainty equals aleatoric plus epistemic uncertainty, and that when the scoring rule is a kernel score, properties of the kernel transfer to the resulting uncertainty measures. Specifically, translation-invariant kernels give translation-invariant measures; translation-invariant kernels convex in one argument make aleatoric uncertainty nondecreasing under convex-order spread; bounded kernels make the influence function bounded, hence robust to outliers; and the energy kernel is the only homogeneous translation-invariant kernel score on the target space, so affine rescaling of data can

What carries the argument

The engine is the kernel score, defined for a continuous negative definite nonnegative kernel by S_k(P,y) = ∫k(x,y)dP(x) − ½∫∫k(x,x')dP(x)dP(x') − ½k(y,y). Its associated entropy and divergence, the latter being the squared maximum mean discrepancy, plug into two estimators—one comparing each predictive distribution to the Bayesian model average, one comparing all pairs—both satisfying total = aleatoric + epistemic. The kernel's properties (boundedness, translation invariance, convexity in one argument, homogeneity) are shown to be inherited by these estimators, which is what turns kernel choice into a design rule for uncertainty quantification.

Load-bearing premise

Two load-bearing premises: the pairwise estimator's monotonicity needs the map from distribution to score to be convex for fixed second argument, an assumption introduced only inside the proof and false for some proper scoring rules; and the kernel-property-to-behavior transfers only hold when the kernel meets the specific convexity or boundedness conditions, which the energy score with exponent below one does not.

What would settle it

For the energy kernel with exponent below one (strictly proper but not convex), construct two second-order distributions ordered by convex order on Gaussian mixture components and compute the pairwise epistemic uncertainty for both; if the less-spread one gives higher epistemic uncertainty, Proposition 5.1(2) as stated for any proper scoring rule is false. The robustness claim can be checked by an outlier ensemble member with growing variance: bounded-kernel uncertainty should saturate while squared-error and energy uncertainties diverge.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Bounded-kernel measures, such as the Gaussian kernel score, have bounded influence functions, so a single bad ensemble member can only shift the uncertainty estimate by a finite amount, unlike squared-error or log-score based measures.
  • Translation-invariant convex or energy kernels make aleatoric uncertainty increase when the predictive distribution spreads in the convex order, giving a formal monotonicity guarantee.
  • The energy score is the only homogeneous translation-invariant kernel score, making it the canonical choice when the measure should respond to scale changes covariantly rather than erratically.
  • The framework nests existing variance-, entropy-, and CRPS-type regression measures, so their different behaviors can be compared under a common set of assumptions.
  • Experiments show the Gaussian kernel's bandwidth can be tuned to improve active-learning acquisition, suggesting task-specific selection is possible within one family.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Inference: because the kernel-score divergence equals squared maximum mean discrepancy, epistemic uncertainty inherits two-sample-testing semantics; out-of-distribution detection could be reinterpreted as a kernel two-sample test between ensemble members and the Bayesian model average.
  • Inference: the design-rule perspective suggests an optimization view the paper leaves implicit—one could select the kernel or its bandwidth by minimizing a downstream task loss, as the active-learning experiments hint, rather than by heuristic.
  • Inference: the robustness results concern the influence function for aleatoric uncertainty; a natural extension is to prove finite-sample breakdown-point bounds for bounded-kernel measures, which would strengthen practical guarantees against outliers.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper proposes a unified framework for second-order uncertainty quantification in regression, based on proper scoring rules and specifically kernel scores. It defines BMA and pairwise estimators for total, aleatoric, and epistemic uncertainty, and proves that the additive decomposition TU=EU+AU holds. The main theoretical results are: (i) Proposition 5.1, claiming that EU is monotone under second-order convex order for any proper scoring rule; (ii) Proposition 5.2, claiming that AU is monotone under first-order convex order for kernel scores with convex translation-invariant kernels; and (iii) Proposition 5.3, showing that bounded kernels yield bounded influence functions and hence robustness. Closed forms are derived for Gaussian and mixture distributions, and experiments on weather post-processing and UCI benchmarks illustrate the qualitative behavior, robustness, and task-adaptation of measures. The manuscript includes public code and emphasizes reproducibility.

Significance. If the theoretical correspondences hold as stated, the framework is a valuable unification of regression uncertainty measures (variance, CRPS/energy, Gaussian-kernel, log-score) and provides a principled design axis through kernel choice. The paper's strengths include explicit closed-form derivations, a clear TU/EU/AU decomposition, a rigorous robustness analysis for bounded kernels, and reproducible experiments. The main risk is that some theorem statements claim more than the proofs establish, particularly the universality of Proposition 5.1 and the energy-score claim in Proposition 5.2. The empirical task-adaptation section also needs clarification regarding how the kernel bandwidth is selected.

major comments (3)
  1. [Section 5, Proposition 5.1 and Appendix A.1] The proposition claims monotonicity of EU under second-order convex order for 'any proper scoring rule'. The BMA part of the proof argues that 'S(P,Q) is affine in Q and therefore convex', but the necessary convexity is in the first argument, P↦S(P,P̄) for fixed P̄; the fact cited is about the second argument and does not imply it. The pairwise part explicitly introduces a new assumption inside the proof ('we require the additional assumption that for a fixed Q, the map P↦S(P,Q) is convex'), an assumption absent from the proposition and not satisfied by every proper scoring rule. Thus the advertised behavior is not a consequence of properness alone; the proposition must be restated with an explicit convexity hypothesis or restricted to the class of kernel scores for which it is verified.
  2. [Section 5, energy score paragraph] The text asserts that the energy score with β∈(0,2) fulfills Proposition 5.2. Proposition 5.2 assumes the translation-invariant kernel ψ(t)=||t||^β is convex in one argument. This holds only for β≥1; for β∈(0,1) the kernel is not convex and the proof's construction of the convex functions φ_{P1}, φ_{P2} breaks down. Please either restrict the energy-score claim to β∈[1,2) or supply a separate proof for 0<β<1.
  3. [Section 6.3, active learning experiment] The experiment varies γ∈(0,2] and reports test CRPS for each value, concluding that 'systematic task adaption' occurs. As presented, this is a sensitivity analysis performed directly on the test objective; no validation-based selection of γ is described. To support the claim that the framework enables task-specific selection, the authors should state a tuning protocol (e.g., a validation split) and evaluate the selected γ on held-out data, or explicitly frame Figure 4 as an exploratory sensitivity check rather than a demonstration of an adaptation procedure.
minor comments (4)
  1. [Definition 4.1 and Section 5] The definition assumes a nonnegative kernel k≥0, but the Gaussian kernel is defined with k(x,y)=-exp(-||x-y||^2/γ^2)<0. Adding a constant to a conditionally negative definite kernel leaves the kernel score unchanged; please state this explicitly to resolve the sign convention.
  2. [Appendix A.1, pairwise proof] The sentence 'Since both sides coincide (by Fubini's theorem)' is inaccurate. The two intermediate terms coincide; the desired inequality follows by chaining E_{Q1,Q1}F ≤ E_{Q1,Q2}F ≤ E_{Q2,Q2}F.
  3. [Tables 1 and 2] Some entries contain formatting artifacts (e.g., '3.49+e04', '1.37+e04'); please clean up the exponent notation.
  4. [Figure 6] Color scales vary across the λ panels, making visual comparison of AU and EU across λ difficult. Consider using common color scales or explicit colorbar limits.

Circularity Check

0 steps flagged

No significant circularity; the central results are derived from the stated kernel-score assumptions rather than fitted or defined into existence.

full rationale

The paper's central claim is that the choice of kernel in a kernel score controls the behavior of the resulting uncertainty measures. This is a derivation, not a fit: TU/EU/AU are defined in Eqs. (4)-(5) as expectations of S, D, and H under a second-order distribution, and Propositions 5.1-5.3 are proved in Appendix A from the stated assumptions (properness, convex order, translation invariance/convexity of the kernel, boundedness). The additive decomposition TU = EU + AU follows from the definitions and is not used as an empirical prediction. No parameter is fitted to data and then renamed a prediction; the gamma-selection experiment in Section 6.3 is a hyperparameter search on the task loss, not an out-of-sample theoretical prediction. The only self-citations (Bülte et al. 2025a, 2025b) are for experimental setup and comparison, and the uniqueness/homogeneity statement about the energy score cites external work (Waghmare & Ziegel 2025). The proof-hypothesis mismatches noted by the skeptic, such as the unstated convexity assumption in the pairwise part of Proposition 5.1(2) and the beta-in-(0,2) versus convexity issue in the energy-score example, are correctness/rigor concerns rather than circularity, because the propositions are not equivalent to their own assumptions. Overall, the derivation is self-contained and does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

2 free parameters · 5 axioms · 0 invented entities

The central construction introduces no new physical entities, but it introduces a family of uncertainty measures with a free kernel choice. The active-learning bandwidth is the main fitted quantity; the energy exponent is a free design knob. The mathematical axioms are mostly standard scoring-rule facts; the convexity assumption for the pairwise estimator is an unflagged ad hoc addition. No invented entities.

free parameters (2)
  • Gaussian kernel bandwidth gamma = median heuristic in most experiments; tuned over (0,2] in active learning (Figure 4)
    S_k_gamma and all Gaussian-kernel uncertainty measures depend on gamma through the closed forms in Eq. (18)-(19); the framework does not derive gamma, and the active-learning 'task adaptation' gain is obtained by selecting gamma on the test objective.
  • Energy score exponent beta = beta=1 (CRPS) in the experiments shown
    The family covers beta in (0,2), but Prop 5.2 is proved only for convex kernels, which requires beta>=1 for ||x-x'||^beta; choosing beta is a free design decision that changes the uncertainty measure.
axioms (5)
  • standard math Proper scoring rule entropy-divergence representation: H(P)=S(P,P), D(P,Q)=S(P,Q)-H(Q), with H concave and D nonnegative.
    Used to define AU, EU, TU in Eqs. (4)-(5); cited to Dawid (2007).
  • domain assumption Convex order <=_cx and second-order convex order <=2_cx are the correct formalization of 'more variability implies more uncertainty'.
    Props 5.1 and 5.2 interpret higher convex-order as higher uncertainty; this is a modeling choice, not derived.
  • standard math Kernel scores with conditionally negative definite kernels are strictly proper and the kernel divergence equals MMD^2.
    Foundation of the whole framework; cited to Waghmare & Ziegel (2025), Gneiting & Raftery (2007), Gretton et al. (2012).
  • ad hoc to paper For the pairwise estimator, P -> S(P,Q) is convex for fixed Q.
    Needed in the proof of Prop 5.1(2) pairwise case (Appendix A.1), but absent from the proposition statement and not satisfied by all proper scoring rules.
  • domain assumption First-order distributions admit densities and finite kernel entropy H_k(P) (P in P_k).
    Section 2 assumes absolute continuity with respect to Lebesgue; Section 4 restricts to P_k; needed for all expectations to exist.

pith-pipeline@v1.3.0-alltime-deepseek · 23949 in / 13886 out tokens · 130722 ms · 2026-08-04T07:25:53.478289+00:00 · methodology

0 comments
read the original abstract

Regression tasks, notably in safety-critical domains, require reliable uncertainty quantification, yet the literature remains largely classification-focused. To address this, we introduce a family of measures for total, aleatoric, and epistemic uncertainty in multivariate regression based on strictly proper kernel scores. The framework provides a principled recipe for designing new uncertainty measures whose behavior, such as tail sensitivity or out-of-distribution responsiveness, is governed by the choice of the underlying kernel, while also encompassing existing measures under a joint analysis. We prove explicit correspondences between properties of the kernel and behavior of resulting uncertainty measures, yielding concrete design guidelines for practitioners. Extensive experiments across structured regression tasks, including spatial and functional domains, demonstrate effectiveness on downstream tasks such as out-of-distribution detection and active learning, and reveal that different kernel choices lead to distinct trade-offs, offering practitioners guidance for task-specific selection.

Figures

Figures reproduced from arXiv: 2510.25599 by Christopher B\"ulte, Eyke H\"ullermeier, Gitta Kutyniok, Yusuf Sale.

Figure 1
Figure 1. Figure 1: Illustration of epistemic uncertainty for a two [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The figure shows AU and EU averaged over a test set of 365 days for the different uncertainty measures. For [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Different task losses (each plot) sorted by each of the different uncertainty measures from highest to lowest [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Continuous ranked probability score with increasing training instances for different model runs with the [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The figure shows the spatial domain used for the distributional regression networks, as well as the corre [PITH_FULL_IMAGE:figures/full_fig_p019_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The figure shows AU and EU averaged over a test set of 365 days for the different uncertainty measures [PITH_FULL_IMAGE:figures/full_fig_p020_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Effect of the added noise δ on the different (aleatoric) uncertainty measures for an ensemble of size M = 25 across all three datasets. The reported values are the mean absolute percentage error from the corresponding measure for the base ensemble. 22 [PITH_FULL_IMAGE:figures/full_fig_p022_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The figure shows the different task losses (each plot) sorted by each of the different uncertainty measures [PITH_FULL_IMAGE:figures/full_fig_p023_8.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. On the QUEST for Uncertainty Quantification via Highest Density Regions

    cs.LG 2026-06 unverdicted novelty 6.0

    QUEST measures uncertainty via the Lebesgue volume of highest-density regions of a distribution's support, evaluated at robustness parameter alpha, and claims to satisfy UQ axioms while outperforming variance and diff...

Reference graph

Works this paper leans on

65 extracted references · 5 canonical work pages · cited by 1 Pith paper

  1. [1]

    Understanding the bias-variance tradeoff of bregman divergences, 2022

    Ben Adlam, Neha Gupta, Zelda Mariet, and Jamie Smith. Understanding the bias-variance tradeoff of bregman divergences, 2022. URL https://arxiv.org/abs/2202.04167

    Pith/arXiv arXiv 2022

  2. [2]

    Andersson, Jacklynn Stott, Remi Lam, Matthew Willson, Alvaro Sanchez-Gonzalez, and Peter Battaglia

    Ferran Alet, Ilan Price, Andrew El-Kadi, Dominic Masters, Stratis Markou, Tom R. Andersson, Jacklynn Stott, Remi Lam, Matthew Willson, Alvaro Sanchez-Gonzalez, and Peter Battaglia. Skillful joint probabilistic weather forecasting from marginals, 2025. URL https://arxiv.org/abs/2506.10772

    Pith/arXiv arXiv 2025

  3. [3]

    Evaluating forecasts for high-impact events using transformed kernel scores

    Sam Allen, David Ginsbourger, and Johanna Ziegel. Evaluating forecasts for high-impact events using transformed kernel scores. SIAM/ASA Journal on Uncertainty Quantification, 11 0 (3): 0 906--940, 2023. doi:10.1137/22M1532184. URL https://doi.org/10.1137/22M1532184

    work page doi:10.1137/22m1532184 2023

  4. [4]

    Deep evidential regression

    Alexander Amini, Wilko Schwarting, Ava Soleimany, and Daniela Rus. Deep evidential regression. In Proceedings of the 34th International Conference on Neural Information Processing Systems, NIPS '20, Red Hook, NY, USA, 2020. Curran Associates Inc. ISBN 9781713829546

    2020

  5. [5]

    Normalizing Flow Ensembles for Rich Aleatoric and Epistemic Uncertainty Modeling

    Lucas Berry and David Meger. Normalizing Flow Ensembles for Rich Aleatoric and Epistemic Uncertainty Modeling . Proceedings of the AAAI Conference on Artificial Intelligence, 37 0 (6): 0 6806--6814, June 2023. ISSN 2374-3468. doi:10.1609/aaai.v37i6.25834

    work page doi:10.1609/aaai.v37i6.25834 2023

  6. [6]

    Efficient epistemic uncertainty estimation in regression ensemble models using pairwise-distance estimators, 2024

    Lucas Berry and David Meger. Efficient epistemic uncertainty estimation in regression ensemble models using pairwise-distance estimators, 2024. URL https://arxiv.org/abs/2308.13498

    arXiv 2024

  7. [7]

    Christopher M. Bishop. Mixture density networks. Workingpaper, Aston University, 1994

    1994

  8. [8]

    Convex Optimization

    Stephen Boyd and Lieven Vandenberghe. Convex Optimization. Cambridge University Press, 2004

    2004

  9. [9]

    Uncertainty quantification for data-driven weather models

    Christopher Bülte, Nina Horat, Julian Quinting, and Sebastian Lerch. Uncertainty quantification for data-driven weather models. Artificial Intelligence for the Earth Systems, 2025 a . doi:10.1175/AIES-D-24-0049.1. URL https://journals.ametsoc.org/view/journals/aies/aop/AIES-D-24-0049.1/AIES-D-24-0049.1.xml

    doi:10.1175/aies-d-24-0049.1 2025

  10. [10]

    An axiomatic assessment of entropy- and variance-based uncertainty quantification in regression, 2025 b

    Christopher Bülte, Yusuf Sale, Timo Löhr, Paul Hofman, Gitta Kutyniok, and Eyke Hüllermeier. An axiomatic assessment of entropy- and variance-based uncertainty quantification in regression, 2025 b . URL https://arxiv.org/abs/2504.18433

    Pith/arXiv arXiv 2025

  11. [11]

    A. P. Dawid. The geometry of proper scoring rules. Annals of the Institute of Statistical Mathematics, 59 0 (1): 0 77--93, February 2007. ISSN 0020-3157, 1572-9052. doi:10.1007/s10463-006-0099-8

    doi:10.1007/s10463-006-0099-8 2007

  12. [12]

    Demaeyer, J

    J. Demaeyer, J. Bhend, S. Lerch, C. Primo, B. Van Schaeybroeck, A. Atencia, Z. Ben Bouall\`egue, J. Chen, M. Dabernig, G. Evans, J. Faganeli Pucer, B. Hooper, N. Horat, D. Jobst, J. Mer s e, P. Mlakar, A. M\"oller, O. Mestre, M. Taillardat, and S. Vannitsem. The euppbench postprocessing benchmark dataset v1.0. Earth System Science Data, 15 0 (6): 0 2635--...

    work page doi:10.5194/essd-15-2635-2023 2023

  13. [13]

    Uncertainty quantification in deep mri reconstruction

    Vineet Edupuganti, Morteza Mardani, and Shreyas Vasanawala. Uncertainty quantification in deep mri reconstruction. IEEE transactions on medical imaging, PP, 09 2020. doi:10.1109/TMI.2020.3025065

    arXiv 2020

  14. [14]

    Graph neural networks and spatial information learning for post-processing ensemble weather forecasts, 2024

    Moritz Feik, Sebastian Lerch, and Jan Stühmer. Graph neural networks and spatial information learning for post-processing ensemble weather forecasts, 2024. URL https://arxiv.org/abs/2407.11050

    Pith/arXiv arXiv 2024

  15. [15]

    Deep B ayesian active learning with image data

    Yarin Gal, Riashat Islam, and Zoubin Ghahramani. Deep B ayesian active learning with image data. In Doina Precup and Yee Whye Teh (eds.), Proceedings of the 34th International Conference on Machine Learning, volume 70 of Proceedings of Machine Learning Research, pp.\ 1183--1192. PMLR, 06--11 Aug 2017. URL https://proceedings.mlr.press/v70/gal17a.html

    2017

  16. [16]

    Tilmann Gneiting and Adrian E. Raftery. Strictly proper scoring rules, prediction, and estimation. Journal of the American Statistical Association, 102 0 (477): 0 359--378, 2007. ISSN 0162-1459. doi:10.1198/016214506000001437

    doi:10.1198/016214506000001437 2007

  17. [17]

    Borgwardt, Malte J

    Arthur Gretton, Karsten M. Borgwardt, Malte J. Rasch, Bernhard Sch \"o lkopf, and Alexander Smola. A kernel two-sample test. The Journal of Machine Learning Research, 13 0 (null): 0 723--773, March 2012. ISSN 1532-4435

    2012

  18. [18]

    Gruber and Florian Buettner

    Sebastian G. Gruber and Florian Buettner. Uncertainty estimates of predictions via a general bias-variance decomposition, 2023. URL https://arxiv.org/abs/2210.12256

    Pith/arXiv arXiv 2023

  19. [19]

    Robust Statistics: The Approach Based on Influence Functions

    Frank Hampel, Elvezio Ronchetti, Peter Rousseeuw, and Werner Stahel. Robust Statistics: The Approach Based on Influence Functions. 03 1986. ISBN 9780471735779. doi:10.1002/9781118186435

    doi:10.1002/9781118186435 1986

  20. [20]

    José Miguel Hernández-Lobato and Ryan P. Adams. Probabilistic backpropagation for scalable learning of bayesian neural networks, 2015. URL https://arxiv.org/abs/1502.05336

    Pith/arXiv arXiv 2015

  21. [21]

    Hans Hersbach, Bill Bell, Paul Berrisford, Shoji Hirahara, Andr \'a s Hor \'a nyi, Joaqu \'i n Mu \ n oz-Sabater, Julien Nicolas, Carole Peubey, Raluca Radu, Dinand Schepers, Adrian Simmons, Cornel Soci, Saleh Abdalla, Xavier Abellan, Gianpaolo Balsamo, Peter Bechtold, Gionata Biavati, Jean Bidlot, Massimo Bonavita, Giovanna De Chiara, Per Dahlgren, Dick ...

    1999

  22. [22]

    Quantifying aleatoric and epistemic uncertainty: A credal approach

    Paul Hofman, Yusuf Sale, and Eyke H \"u llermeier. Quantifying aleatoric and epistemic uncertainty: A credal approach. In ICML 2024 Workshop on Structured Probabilistic Inference & Generative Modeling , 2024 a . URL https://openreview.net/forum?id=MhLnSoWp3p

    2024

  23. [23]

    Quantifying aleatoric and epistemic uncertainty with proper scoring rules, 2024 b

    Paul Hofman, Yusuf Sale, and Eyke Hüllermeier. Quantifying aleatoric and epistemic uncertainty with proper scoring rules, 2024 b . URL https://arxiv.org/abs/2404.12215

    Pith/arXiv arXiv 2024

  24. [24]

    Bayesian active learning for classification and preference learning

    Neil Houlsby, Ferenc Husz \'a r, Zoubin Ghahramani, and M \'a t \'e Lengyel. Bayesian active learning for classification and preference learning. arXiv preprint arXiv:1112.5745, 2011

    Pith/arXiv arXiv 2011

  25. [25]

    Aleatoric and epistemic uncertainty in machine learning: An introduction to concepts and methods

    Eyke H \"u llermeier and Willem Waegeman. Aleatoric and epistemic uncertainty in machine learning: An introduction to concepts and methods. Machine learning, 110 0 (3): 0 457--506, 2021

    2021

  26. [26]

    Kelen, \'A d \'a m Jung, P \'e ter Kersch, and Andras A Benczur

    Domokos M. Kelen, \'A d \'a m Jung, P \'e ter Kersch, and Andras A Benczur. Distribution-free data uncertainty for neural network regression. In The Thirteenth International Conference on Learning Representations, 2025. URL https://openreview.net/forum?id=pDDODPtpx9

    2025

  27. [27]

    Batchbald: Efficient and diverse batch acquisition for deep bayesian active learning

    Andreas Kirsch, Joost van Amersfoort, and Yarin Gal. Batchbald: Efficient and diverse batch acquisition for deep bayesian active learning. In H. Wallach, H. Larochelle, A. Beygelzimer, F. d Alch\' e -Buc, E. Fox, and R. Garnett (eds.), Advances in Neural Information Processing Systems, volume 32. Curran Associates, Inc., 2019. URL https://proceedings.neur...

    arXiv 2019

  28. [28]

    Rage Against the Mean -- A Review of Distributional Regression Approaches

    Thomas Kneib, Alexander Silbersdorff, and Benjamin S \"a fken. Rage Against the Mean -- A Review of Distributional Regression Approaches . Econometrics and Statistics, 26: 0 99--123, April 2023. ISSN 2452-3062. doi:10.1016/j.ecosta.2021.07.006

    doi:10.1016/j.ecosta.2021.07.006 2023

  29. [29]

    From risk to uncertainty: Generating predictive uncertainty measures via bayesian estimation

    Nikita Kotelevskii, Vladimir Kondratyev, Martin Tak \'a c , Eric Moulines, and Maxim Panov. From risk to uncertainty: Generating predictive uncertainty measures via bayesian estimation. In The Thirteenth International Conference on Learning Representations, 2025. URL https://openreview.net/forum?id=cWfpt2t37q

    2025

  30. [30]

    Nonparametric uncertainty quantification for single deterministic neural network

    Nikita Yurevich Kotelevskii, Aleksandr Artemenkov, Kirill Fedyanin, Fedor Noskov, Alexander Fishkov, Artem Shelmanov, Artem Vazhentsev, Aleksandr Petiushko, and Maxim Panov. Nonparametric uncertainty quantification for single deterministic neural network. In Alice H. Oh, Alekh Agarwal, Danielle Belgrave, and Kyunghyun Cho (eds.), Advances in Neural Inform...

    2022

  31. [31]

    Simple and scalable predictive uncertainty estimation using deep ensembles

    Balaji Lakshminarayanan, Alexander Pritzel, and Charles Blundell. Simple and scalable predictive uncertainty estimation using deep ensembles. In I. Guyon, U. Von Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan, and R. Garnett (eds.), Advances in Neural Information Processing Systems, volume 30. Curran Associates, Inc., 2017. URL https://proceed...

    2017

  32. [32]

    Stewart, Stefan Depeweg, and Eric Nalisnick

    Nils Lehmann, Nina Maria Gottschling, Jakob Gawlikowski, Adam J. Stewart, Stefan Depeweg, and Eric Nalisnick. Lightning uq box: Uncertainty quantification for neural networks. Journal of Machine Learning Research, 26 0 (54): 0 1--7, 2025. URL http://jmlr.org/papers/v26/24-2110.html

    2025

  33. [33]

    The enterprise of knowledge: An essay on knowledge, credal probability, and chance

    Isaac Levi. The enterprise of knowledge: An essay on knowledge, credal probability, and chance. MIT press, 1980

    1980

  34. [34]

    Text Classification using String Kernels

    Huma Lodhi, Craig Saunders, John Shawe-Taylor , Nello Cristianini, and Chris Watkins. Text Classification using String Kernels . Journal of Machine Learning Research, 2 0 (Feb): 0 419--444, 2002. ISSN ISSN 1533-7928

    2002

  35. [35]

    o hr, Michael Ingrisch, and Eyke H \

    Timo L \"o hr, Michael Ingrisch, and Eyke H \"u llermeier. Towards aleatoric and epistemic uncertainty in medical image classification. In International Conference on Artificial Intelligence in Medicine, pp.\ 145--155. Springer, 2024

    2024

  36. [36]

    Predictive uncertainty estimation via prior networks

    Andrey Malinin and Mark Gales. Predictive uncertainty estimation via prior networks. In S. Bengio, H. Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi, and R. Garnett (eds.), Advances in Neural Information Processing Systems, volume 31. Curran Associates, Inc., 2018. URL https://proceedings.neurips.cc/paper_files/paper/2018/file/3ea2db50e62ceefceaf70a9...

    2018

  37. [37]

    Uncertainty estimation in autoregressive structured prediction

    Andrey Malinin and Mark Gales. Uncertainty estimation in autoregressive structured prediction. In International Conference on Learning Representations, 2021. URL https://openreview.net/forum?id=jN5y-zb5Q7m

    2021

  38. [38]

    Regression prior networks, 2020

    Andrey Malinin, Sergey Chervontsev, Ivan Provilkov, and Mark Gales. Regression prior networks, 2020. URL https://arxiv.org/abs/2006.11590

    Pith/arXiv arXiv 2020

  39. [39]

    Multivariate deep evidential regression, 2022

    Nis Meinert and Alexander Lavin. Multivariate deep evidential regression, 2022. URL https://arxiv.org/abs/2104.06135

    Pith/arXiv arXiv 2022

  40. [40]

    Evaluating uncertainty quantification in end-to-end autonomous driving control, 2018

    Rhiannon Michelmore, Marta Kwiatkowska, and Yarin Gal. Evaluating uncertainty quantification in end-to-end autonomous driving control, 2018. URL https://arxiv.org/abs/1811.06817

    Pith/arXiv arXiv 2018

  41. [41]

    Benchmarking uncertainty disentanglement: Specialized uncertainties for specialized tasks

    B\' a lint Mucs\' a nyi, Michael Kirchhof, and Seong Joon Oh. Benchmarking uncertainty disentanglement: Specialized uncertainties for specialized tasks. In A. Globerson, L. Mackey, D. Belgrave, A. Fan, U. Paquet, J. Tomczak, and C. Zhang (eds.), Advances in Neural Information Processing Systems, volume 37, pp.\ 50972--51038. Curran Associates, Inc., 2024....

    2024

  42. [42]

    How to measure uncertainty in uncertainty sampling for active learning

    Vu-Linh Nguyen, Mohammad Shaker, and Eyke Hüllermeier. How to measure uncertainty in uncertainty sampling for active learning. Machine Learning, 111, 01 2022. doi:10.1007/s10994-021-06003-9

    doi:10.1007/s10994-021-06003-9 2022

  43. [43]

    Estimation of entropy and mutual information

    Liam Paninski. Estimation of entropy and mutual information. Neural Comput., 15 0 (6): 0 1191–1253, June 2003. ISSN 0899-7667. doi:10.1162/089976603321780272. URL https://doi.org/10.1162/089976603321780272

    doi:10.1162/089976603321780272 2003

  44. [44]

    P. B. Patnaik. The non-central χ2- and f-distribution and their applications. Biometrika, 36 0 (1/2): 0 202--232, 1949. ISSN 00063444, 14643510. URL http://www.jstor.org/stable/2332542

    arXiv 1949

  45. [45]

    The hidden uncertainty in a neural networks activations, 2021

    Janis Postels, Hermann Blum, Yannick Strümpler, Cesar Cadena, Roland Siegwart, Luc Van Gool, and Federico Tombari. The hidden uncertainty in a neural networks activations, 2021. URL https://arxiv.org/abs/2012.03082

    Pith/arXiv arXiv 2021

  46. [46]

    Andersson, Andrew El-Kadi , Dominic Masters, Timo Ewalds, Jacklynn Stott, Shakir Mohamed, Peter Battaglia, Remi Lam, and Matthew Willson

    Ilan Price, Alvaro Sanchez-Gonzalez , Ferran Alet, Tom R. Andersson, Andrew El-Kadi , Dominic Masters, Timo Ewalds, Jacklynn Stott, Shakir Mohamed, Peter Battaglia, Remi Lam, and Matthew Willson. Probabilistic weather forecasting with machine learning. Nature, 637 0 (8044): 0 84--90, January 2025. ISSN 1476-4687. doi:10.1038/s41586-024-08252-9

    doi:10.1038/s41586-024-08252-9 2025

  47. [47]

    Neural networks for postprocessing ensemble weather forecasts

    Stephan Rasp and Sebastian Lerch. Neural networks for postprocessing ensemble weather forecasts. Monthly Weather Review, 146 0 (11): 0 3885--3900, 2018. ISSN 0027-0644. doi:10.1175/MWR-D-18-0187.1

    doi:10.1175/mwr-d-18-0187.1 2018

  48. [48]

    Weatherbench 2: A benchmark for the next generation of data-driven global weather models, 2024

    Stephan Rasp, Stephan Hoyer, Alexander Merose, Ian Langmore, Peter Battaglia, Tyler Russel, Alvaro Sanchez-Gonzalez, Vivian Yang, Rob Carver, Shreya Agrawal, Matthew Chantry, Zied Ben Bouallegue, Peter Dueben, Carla Bromberg, Jared Sisk, Luke Barrington, Aaron Bell, and Fei Sha. Weatherbench 2: A benchmark for the next generation of data-driven global wea...

    Pith/arXiv arXiv 2024

  49. [49]

    Second-order uncertainty quantification: A distance-based approach, 2023 a

    Yusuf Sale, Viktor Bengs, Michele Caprio, and Eyke Hüllermeier. Second-order uncertainty quantification: A distance-based approach, 2023 a . URL https://arxiv.org/abs/2312.00995

    Pith/arXiv arXiv 2023

  50. [50]

    Second-order uncertainty quantification: Variance-based measures, 2023 b

    Yusuf Sale, Paul Hofman, Lisa Wimmer, Eyke Hüllermeier, and Thomas Nagler. Second-order uncertainty quantification: Variance-based measures, 2023 b . URL https://arxiv.org/abs/2401.00276

    Pith/arXiv arXiv 2023

  51. [51]

    Michael Scheuerer and Thomas M. Hamill. Variogram-based proper scoring rules for probabilistic forecasts of multivariate quantities. Monthly Weather Review, 143 0 (4): 0 1321 -- 1334, 2015. doi:10.1175/MWR-D-14-00269.1. URL https://journals.ametsoc.org/view/journals/mwre/143/4/mwr-d-14-00269.1.xml

    work page doi:10.1175/mwr-d-14-00269.1 2015

  52. [52]

    Introducing an improved information-theoretic measure of predictive uncertainty, 2023

    Kajetan Schweighofer, Lukas Aichberger, Mykyta Ielanskyi, and Sepp Hochreiter. Introducing an improved information-theoretic measure of predictive uncertainty, 2023. URL https://arxiv.org/abs/2311.08309

    Pith/arXiv arXiv 2023

  53. [53]

    Equivalence of distance-based and RKHS-based statistics in hypothesis testing

    Dino Sejdinovic, Bharath Sriperumbudur, Arthur Gretton, and Kenji Fukumizu. Equivalence of distance-based and RKHS-based statistics in hypothesis testing. The Annals of Statistics, 41 0 (5): 0 2263--2291, October 2013. ISSN 0090-5364, 2168-8966. doi:10.1214/13-AOS1140

    doi:10.1214/13-aos1140 2013

  54. [54]

    Shanthikumar

    Moshe Shaked and J. Shanthikumar. Stochastic Orders. 01 2007. ISBN 978-0-387-32915-4. doi:10.1007/978-0-387-34675-5

    doi:10.1007/978-0-387-34675-5 2007

  55. [55]

    Székely and Maria L

    Gábor J. Székely and Maria L. Rizzo. Energy statistics: A class of statistics based on distances. Journal of Statistical Planning and Inference, 143 0 (8): 0 1249--1272, 2013. ISSN 0378-3758. doi:https://doi.org/10.1016/j.jspi.2013.03.018. URL https://www.sciencedirect.com/science/article/pii/S0378375813000633

    doi:10.1016/j.jspi.2013.03.018 2013

  56. [56]

    A Deeper Look into Aleatoric and Epistemic Uncertainty Disentanglement

    Matias Valdenegro-Toro and Daniel Saromo Mori. A Deeper Look into Aleatoric and Epistemic Uncertainty Disentanglement . In 2022 IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops (CVPRW), pp.\ 1508--1516, Los Alamitos, CA, USA, June 2022. IEEE Computer Society. doi:10.1109/CVPRW56347.2022.00157. URL https://doi.ieeecomputersociety.or...

    arXiv 2022

  57. [57]

    Vishwanathan, Nicol N

    S.V.N. Vishwanathan, Nicol N. Schraudolph, Risi Kondor, and Karsten M. Borgwardt. Graph kernels. Journal of Machine Learning Research, 11 0 (40): 0 1201--1242, 2010. URL http://jmlr.org/papers/v11/vishwanathan10a.html

    2010

  58. [58]

    Proper scoring rules for estimation and forecast evaluation, 2025

    Kartik Waghmare and Johanna Ziegel. Proper scoring rules for estimation and forecast evaluation, 2025. URL https://arxiv.org/abs/2504.01781

    Pith/arXiv arXiv 2025

  59. [59]

    Quantifying aleatoric and epistemic uncertainty in machine learning: Are conditional entropy and mutual information appropriate measures? In Robin J

    Lisa Wimmer, Yusuf Sale, Paul Hofman, Bernd Bischl, and Eyke H\"ullermeier. Quantifying aleatoric and epistemic uncertainty in machine learning: Are conditional entropy and mutual information appropriate measures? In Robin J. Evans and Ilya Shpitser (eds.), Proceedings of the Thirty-Ninth Conference on Uncertainty in Artificial Intelligence, volume 216 of...

    2023

  60. [60]

    Moments and absolute moments of the normal distribution, 2014

    Andreas Winkelbauer. Moments and absolute moments of the normal distribution, 2014. URL https://arxiv.org/abs/1209.4340

    Pith/arXiv arXiv 2014

  61. [61]

    Julia Wolleb, Robin Sandkühler, Florentin Bieder, Philippe Valmaggia, and Philippe C. Cattin. Diffusion models for implicit image segmentation ensembles, 2021. URL https://arxiv.org/abs/2112.03145

    Pith/arXiv arXiv 2021

  62. [62]

    George Wynne and Andrew B. Duncan. A kernel two-sample test for functional data. Journal of Machine Learning Research, 23 0 (73): 0 1--51, 2022. URL http://jmlr.org/papers/v23/20-1180.html

    2022

  63. [63]

    Characteristic kernels on Hilbert spaces, Banach spaces, and on sets of measures

    Johanna Ziegel, David Ginsbourger, and Lutz D \"u mbgen. Characteristic kernels on Hilbert spaces, Banach spaces, and on sets of measures. Bernoulli, 30 0 (2): 0 1441--1457, May 2024. ISSN 1350-7265. doi:10.3150/23-BEJ1639

    work page doi:10.3150/23-bej1639 2024

  64. [64]

    , " * write output.state after.block = add.period write

    ENTRY address author booktitle chapter doi edition editor eid howpublished institution journal key month note number organization pages publisher school series title type url volume year label INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.all := #1 'mid.sentence := #2 'after.sentence := #3 '...

  65. [65]

    write newline

    " write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in capitalize ":" * " " *...