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arxiv: 2606.10777 · v1 · pith:NNTBXMPQnew · submitted 2026-06-09 · 💻 cs.LG

Can we trust our models? Epistemic calibration in second-order classification

Arthur Hoarau This is my paper

Pith reviewed 2026-06-27 13:55 UTC · model grok-4.3

classification 💻 cs.LG
keywords epistemic calibrationuncertainty estimationsecond-order classificationcalibration errorepistemic uncertaintymachine learning trustworthiness
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The pith

Epistemic calibration requires that a model's reported uncertainty about its uncertainty matches the actual spread of its predictions around the truth.

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

Classical calibration checks whether predicted probabilities match observed frequencies but leaves unexamined whether a model's estimate of its own epistemic uncertainty is reliable. This paper defines epistemic calibration as the requirement that reported epistemic uncertainty faithfully reflects the dispersion of model predictions around the ground truth. It establishes that this criterion is strictly stronger than classical calibration and detects failure modes invisible to standard metrics. The paper also states an impossibility theorem that holds under the epistemic calibration hypothesis and introduces the Expected Epistemic Calibration Error as a consistent estimator of the underlying True Epistemic Calibration Error.

Core claim

Epistemic calibration measures whether reported epistemic uncertainty faithfully reflects the dispersion of model predictions around the ground truth. It is a strictly stronger notion than classical calibration and captures failure modes invisible to standard metrics. An impossibility theorem relates the concept to existing literature under the epistemic calibration hypothesis. The Expected Epistemic Calibration Error (EECE) is a consistent estimator of the True Epistemic Calibration Error (TECE). Experiments across uncertainty quantification methods confirm that the criterion is coherent and reveals substantial differences among methods despite similar predictive performance.

What carries the argument

Epistemic calibration, the alignment between a model's reported epistemic uncertainty and the observed dispersion of its predictions around ground truth.

If this is right

  • Epistemic calibration captures failure modes invisible to classical calibration metrics.
  • An impossibility theorem holds for methods satisfying the epistemic calibration hypothesis.
  • EECE serves as a consistent estimator of TECE.
  • Uncertainty quantification methods can differ substantially in epistemic calibration even when predictive performance is comparable.

Where Pith is reading between the lines

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

  • Models selected for high-stakes use might change if epistemic calibration replaces or supplements classical calibration as the audit criterion.
  • The same comparison between reported uncertainty and prediction dispersion could be examined in regression or other output types beyond classification.
  • Training procedures for second-order models might be adjusted to optimize directly for low epistemic calibration error.

Load-bearing premise

Reported epistemic uncertainty can be meaningfully compared to the dispersion of model predictions around the ground truth as the target quantity for trustworthiness.

What would settle it

A second-order model with low EECE yet systematic mismatch between reported epistemic uncertainty and the actual variance of predictions around ground truth would show that EECE does not consistently estimate TECE.

Figures

Figures reproduced from arXiv: 2606.10777 by Arthur Hoarau.

Figure 1
Figure 1. Figure 1: 1a: Calibration plot with bins vs. Random Forest. Regularization on depth or leaf density improves cali￾bration. 1b: Toy Dataset used in the following to illustrate calibration and epistemic calibration. Two-class partially overlapping Gaussian kernels. 1c: Smooth calibration curve through Kernel Density Estimation for an almost perfectly calibrated Bayesian Logistic Regression. In the literature, two prim… view at source ↗
Figure 2
Figure 2. Figure 2: For each model, the left panel displays the estimated epistemic uncertainty [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: 3a: EECE for different models vs. Toy dataset. 3b: Random Forest’s EECE behavior vs. increasing noise (left) and training set size (rigt) or Toy dataset. (a) (b) [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: 4a: ECE and EECE for multiple uncerainty quantification methods on CIFAR-10 under aggregated One [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Absolute difference between the estimator through binning and the true expected value [PITH_FULL_IMAGE:figures/full_fig_p017_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: For each model, the left panel displays the estimated epistemic uncertainty [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗
read the original abstract

Uncertainty estimation is critical for deploying machine learning models in high-stakes settings. However, classical calibration only assesses the reliability of predicted probabilities and does not evaluate whether epistemic uncertainty estimates are themselves trustworthy. This limitation is particularly relevant for second-order classification models. We introduce epistemic calibration, a principled criterion that measures whether reported epistemic uncertainty faithfully reflects the dispersion of model predictions around the ground truth. We show that epistemic calibration is a strictly stronger notion than classical calibration and captures failure modes invisible to standard metrics. We relate this work to the existing literature through an impossibility theorem that holds under the epistemic calibration hypothesis. To operationalize this concept, we propose the Expected Epistemic Calibration Error (EECE), which we prove to be a consistent estimator of a True Epistemic Calibration Error (TECE). Experiments across a broad range of uncertainty quantification methods show that epistemic calibration is a coherent and meaningful criterion and reveal substantial differences across methods, despite similar predictive performance.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper introduces epistemic calibration as a new criterion for second-order classification models, measuring whether reported epistemic uncertainty faithfully reflects the dispersion of model predictions around the ground truth. It claims this notion is strictly stronger than classical calibration and captures additional failure modes, presents an impossibility theorem that holds under the epistemic calibration hypothesis, proposes the Expected Epistemic Calibration Error (EECE) as a consistent estimator of the True Epistemic Calibration Error (TECE), and reports experiments across uncertainty quantification methods showing that the criterion is coherent and reveals differences despite similar predictive performance.

Significance. If the proofs of strict dominance, the impossibility theorem, and consistency of EECE hold, the work provides a principled extension of calibration concepts that addresses trustworthiness of epistemic uncertainty estimates in high-stakes settings. The consistency result for EECE and the experimental demonstration of method differences are concrete strengths that could influence evaluation practices in uncertainty quantification.

major comments (2)
  1. [Theory section on impossibility theorem] The impossibility theorem (abstract and theory section) is conditioned on the epistemic calibration hypothesis; the manuscript must explicitly demonstrate that the definitions of TECE and EECE are independent of this hypothesis and of any fitted parameters to confirm the argument is non-circular.
  2. [Section proving consistency of EECE] The claim that EECE is a consistent estimator of TECE (abstract) is central; the full derivation and assumptions (e.g., in the section proving consistency) must be provided in detail, as the abstract alone does not allow verification of the estimator's properties under the stated conditions.
minor comments (2)
  1. [Abstract] The abstract refers to experiments across 'a broad range of uncertainty quantification methods' but does not list them; adding the specific methods and a summary table of results would improve clarity.
  2. [Definition of epistemic calibration] The weakest assumption (comparability of reported epistemic uncertainty to dispersion around ground truth) is definitional for the new concept and does not introduce inconsistency, but the manuscript should state this choice explicitly in the definition section.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which help clarify the presentation of our theoretical contributions. We address each major comment below.

read point-by-point responses

  1. Referee: [Theory section on impossibility theorem] The impossibility theorem (abstract and theory section) is conditioned on the epistemic calibration hypothesis; the manuscript must explicitly demonstrate that the definitions of TECE and EECE are independent of this hypothesis and of any fitted parameters to confirm the argument is non-circular.

    Authors: We agree that an explicit statement is needed to preclude any perception of circularity. The definitions of TECE (Definition 3) and EECE (Definition 4) are introduced in Section 3 solely in terms of the joint distribution of model predictions and ground-truth labels; they make no reference to the epistemic calibration hypothesis or to any fitted parameters. The hypothesis is stated only at the beginning of Section 4, where the impossibility theorem is proved under that assumption. In the revision we will insert a short clarifying paragraph immediately before the theorem stating that the preceding definitions are independent of the hypothesis. revision: yes

  2. Referee: [Section proving consistency of EECE] The claim that EECE is a consistent estimator of TECE (abstract) is central; the full derivation and assumptions (e.g., in the section proving consistency) must be provided in detail, as the abstract alone does not allow verification of the estimator's properties under the stated conditions.

    Authors: The consistency proof, including the full set of assumptions (i.i.d. sampling, finite second moments of the epistemic uncertainty, and the use of the law of large numbers), appears in Section 5 together with the supporting lemmas in the appendix. Nevertheless, to facilitate direct verification from the main text we will expand Section 5 by (i) listing all assumptions in a dedicated box and (ii) inserting the key intermediate steps of the proof that were previously only sketched. This change will be made in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper defines epistemic calibration as a new criterion measuring whether reported epistemic uncertainty reflects dispersion of predictions around ground truth, then derives that it is strictly stronger than classical calibration, presents an impossibility theorem conditional on the epistemic calibration hypothesis, and proves EECE is a consistent estimator of TECE. These steps follow from the introduced definitions and standard consistency arguments without reducing to fitted inputs renamed as predictions, self-citations as load-bearing premises, or any self-definitional equivalence where outputs equal inputs by construction. The conditional nature of the impossibility theorem is a standard logical structure and does not create circularity. Core claims remain independent of the paper's own fitted values or prior self-referential results.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the separation of epistemic uncertainty and the existence of a ground-truth dispersion quantity; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Epistemic uncertainty estimates can be isolated and compared to prediction dispersion around ground truth in second-order models.
    Required for the definition of epistemic calibration and the claim that it is strictly stronger.

pith-pipeline@v0.9.1-grok · 5686 in / 1188 out tokens · 20215 ms · 2026-06-27T13:55:48.723792+00:00 · methodology

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