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Epistemic Uncertainty in Conformal Scores: A Unified Approach

T0 review · 3 major / 5 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read EPICSCORE wraps any conformal score in a Bayesian predictive CDF, keeping finite-sample marginal coverage and adding asymptotic conditional coverage by widening intervals where data are sparse.

desk verdict A useful Bayesian wrapper for conformal scores with honest marginal coverage, but the asymptotic conditional coverage claim leans on a strong assumption the experiments never verify. read the letter →

arxiv 2502.06995 v2 pith:ZL2RFNN6 submitted 2025-02-10 stat.ML cs.LG

Luben M. C. Cabezas , Vagner S. Santos , Thiago R. Ramos , Rafael Izbicki This is my paper · ORCID
classification stat.MLcs.LG MSC 62G15
keywords conformalpredictionepistemicuncertaintyBayesianpredictivedistributionconditionalcoveragedistribution-freeguaranteesGaussianprocessesMonteCarlodropoutscores
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proposes EPICSCORE, a wrapper that takes any conformal score $s(x,y)$ — a number saying how unusual a candidate label $y$ is for features $x$ — and replaces it with $s'(x,y) = F(s(x,y)\mid x,\mathcal{D})$, the cumulative probability of that score under a Bayesian model fitted to calibration data. The claim is that this one transformation injects epistemic uncertainty into any conformal method: prediction intervals widen in data-sparse regions, where the Bayesian posterior is spread out, and stay narrow where data are abundant, while the finite-sample marginal coverage guarantee of conformal prediction survives untouched for every Bayesian model. Under a uniform-convergence assumption on the Bayesian model, the paper further proves asymptotic conditional coverage, the strongest achievable form of per-feature validity. If true, this means practitioners can attach epistemic uncertainty to any existing conformal pipeline using Gaussian processes, BART, or Monte Carlo dropout, and still keep a distribution-free guarantee.

What carries the argument

The load-bearing object is the Bayesian posterior predictive CDF, $F(s\mid x,\mathcal{D}) = \int F(s\mid x,\theta)\,f(\theta\mid\mathcal{D})\,d\theta$, computed from a subset of the calibration data and used to define $s'(x,y) = F(s(x,y)\mid x,\mathcal{D})$. It performs a distributional transform: it ranks the original conformal score against the posterior-averaged distribution of scores at $x$, so the conformal threshold $t_{1-\alpha}$ in the transformed space corresponds to the per-input quantile $F^{-1}(t_{1-\alpha}\mid x,\mathcal{D})$ in the original score space. That per-input quantile is the mechanism that turns 'large posterior spread at $x$' into 'widened interval at $x$'; when the posterior concentrates on the true score distribution, the transform reduces to the estimated-CDF score whose asymptotic conditional coverage is already known.

What would settle it

Simulate a regression problem with a strongly skewed, heavy-tailed conditional score distribution and a clearly defined data-sparse region, then fit EPICSCORE with a normal-likelihood Gaussian process, which cannot converge to that true score distribution. If empirical conditional coverage in the sparse region fails to approach $1-\alpha$ as the calibration sample grows, Assumption 1 is violated and the asymptotic conditional coverage claim fails for misspecified Bayesian models; if it does approach $1-\alpha$, the claim survives this particular misspecification.

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Extended reading notes

Core claim

The central claim is that mapping a conformal score through its Bayesian posterior predictive CDF, $s'(x,y)=F(s(x,y)\mid x,\mathcal{D})$, converts epistemic uncertainty into a learned, feature-dependent cutoff: the prediction region becomes $\{y : s(x,y) \le F^{-1}(t_{1-\alpha}\mid x,\mathcal{D})\}$, so where the Bayesian model's predictive distribution is wide, the effective threshold is automatically looser. Theorem 1 proves the transformed score preserves finite-sample marginal coverage for any Bayesian model, because the exchangeability argument behind split conformal prediction is untouched by the transform. Theorem 2 proves that when the predictive CDF converges uniformly in probability to the true conditional distribution of the score (Assumption 1), the regions attain asymptotic conditional coverage, $P(Y\in R_{\mathrm{EPIC}}(X)\mid X=x) \to 1-\alpha$, recovering in the limit the CDF-transformed conformal score of Dheur et al. (2025).

Load-bearing premise

The load-bearing premise is Assumption 1: the Bayesian model's predictive distribution of the conformal score must converge uniformly in probability to the true score distribution as the fitting sample grows — if the model family is wrong and this convergence fails, the asymptotic conditional coverage claim (Theorem 2) collapses, although the always-valid finite-sample marginal coverage (Theorem 1) remains.

Editorial extensions

If this is right

  • Any conformal score — regression residuals, CQR, APS, or density-based scores — can be wrapped with the same recipe, and the marginal coverage guarantee of Theorem 1 holds regardless of which Bayesian model is used.
  • The effective cutoff $F^{-1}(t_{1-\alpha}\mid x,\mathcal{D})$ is a learned function of the features, so the same calibration quantile produces wide intervals where data are scarce and narrow ones where data are abundant.
  • As the calibration sample grows and the Bayesian model converges to the true score distribution, EPICSCORE inherits asymptotic conditional coverage, a property that standard split conformal prediction lacks.
  • In classification, EPICSCORE equals APS computed with predictive probabilities rather than point estimates, and the CIFAR-100 experiments report prediction sets that expand for outlier images, contract for in-distribution images, and improve size-stratified coverage over plain APS.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The 'epistemic uncertainty' EPICSCORE reports is entirely inherited from the Bayesian model's posterior spread: a misspecified model family still produces adaptive intervals, but the widening may occur in the wrong places, and only the marginal guarantee remains intact.
  • The paper's own prior-concentration experiment shows a diffuse prior widens sparse-region intervals while a concentrated prior regularizes them, which suggests a practical tuning recipe — selecting the Bayesian prior to optimize a coverage-sharpness metric such as AISL — that the paper mentions but does not develop.
  • Because the classification variant produces the same score for both standard APS scoring functions, the adaptivity is carried by the predictive distribution $P(y\mid x,\mathcal{D})$, not by the base score, hinting the wrapper applies equally to other discrete structured-prediction spaces.
  • The paper lists distribution shift as future work; a direct test would check whether the widened sparse-region intervals are exactly the ones that preserve coverage when the test distribution shifts toward those regions.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper introduces EPICSCORE, a wrapper that replaces any conformal score s(x,y) by its Bayesian posterior predictive CDF, s'(x,y)=F(s(x,y)|x,D), where D is a subset of the calibration set. The new score is then used in standard split conformal prediction. The authors prove finite-sample marginal coverage (Theorem 1) and claim asymptotic conditional coverage under a uniform-convergence assumption on the posterior predictive CDF (Theorem 2). They instantiate the method with BART, variational Gaussian processes, and MDNs with MC dropout, and report experiments on 13 regression/quantile-regression benchmarks plus a CIFAR-100 classification study.

Significance. The finite-sample marginal-coverage guarantee is genuine and correctly obtained: the Bayesian model is fitted on Dcal,1 while the conformal quantile is computed on the disjoint set Dcal,2, so the exchangeability argument is not circular. The paper also ships code and uses 50 repeated runs, which is a strength. If the asymptotic conditional-coverage claim were fully established, EPICSCORE would be a useful model-agnostic tool for widening conformal regions in data-sparse areas. However, the asymptotic claim currently rests on Assumption 1, which is an oracle correct-specification and uniform-concentration condition that is not verified for, and is unlikely to hold exactly for, the approximate Bayesian models used in the experiments. The contribution is therefore promising but its headline theoretical property is not yet established at the level claimed.

major comments (3)
  1. [Section 3, Assumption 1 and Theorem 2] The asymptotic conditional-coverage claim is conditional on an assumption that is not satisfied by the algorithm as implemented. MC dropout is a variational approximation rather than exact posterior inference, and the Gaussian, mixture, and BART families are not plausible exact models for absolute-residual or quantile-based conformal scores. The proof requires that F(s(X,Y)|X,theta*) be the exact conditional CDF of the score, because the uniformity step P(s''(X,Y)<=u|X)=u is used. For a misspecified family, s'' is not marginally uniform and the proof collapses. Please either state Theorem 2 explicitly as a correct-specification oracle result and reword the abstract and contributions accordingly, or provide checkable sufficient conditions and empirical validation that the condition holds approximately for at least one instantiation.
  2. [Appendix E, proof of Theorem 2] The proof contains a technical gap in the final probability calculation. The equality P(s''(X,Y)<=t''_{1-alpha}+2*epsilon|X)=t''_{1-alpha}+2*epsilon is not valid when t''_{1-alpha}+2*epsilon>1, since the uniform CDF is capped at 1. Moreover, t''_{1-alpha} is a random variable depending on Dcal,2, so the conditional probability should be E[ min(t''_{1-alpha}+2*epsilon,1) | X ] rather than t''_{1-alpha}+2*epsilon. The lower-bound half of the claimed absolute difference is also not derived. As written, the conclusion |P(s'(X,Y)<=t'_{1-alpha}|X)-t''_{1-alpha}|<=2*epsilon+delta is not proven.
  3. [Section 3, Assumption 1] The 'true parameter' theta* is never formally defined. The conformal score s(x,y) itself depends on Dtrain, so the target F(s|x,theta*) is not a fixed object if Dtrain is allowed to grow; the theorem only sends |Dcal| to infinity, but the paper should state that Dtrain is held fixed and should define theta* relative to that fixed score. In addition, Assumption 1 demands sup-norm uniform convergence of a posterior predictive CDF, which is not a standard consequence of Bayesian consistency for the nonparametric families used, and no theorem or reference establishing it for BART, variational GPs, or MC-dropout MDNs is provided.
minor comments (5)
  1. [Abstract and Section 3] The phrase 'distribution-free guarantees' should be qualified: finite-sample marginal coverage is distribution-free, but the asymptotic conditional-coverage result is not distribution-free and depends on Assumption 1.
  2. [Section 4.3 and Table 3] The claim that EPICSCORE 'consistently achieves higher' outlier-to-inlier interval length ratios is overstated; for example, on bike and cycle the EPIC-MDN ratio is below 1 and below several baselines.
  3. [Section 2.2] The equivalent expression R_EPIC(x)={y: s(x,y)<=F^{-1}(t_{1-alpha}|x,D)} assumes invertibility of the predictive CDF; for discrete scores, as in the classification special case, the direct definition via s'(x,y) should be used.
  4. [Appendix D.5] If the AISL metric is used to tune the BART prior hyperparameter beta, the paper should specify which data split is used for computing AISL, to avoid tuning on the test set.
  5. [Throughout] There are scattered typographical errors, such as 'predtion' in Appendix C.1 and 'comparisson' in Appendix D.6, which should be corrected.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the score transformation is fitted on a separate calibration subset, the coverage quantile is computed on a disjoint subset, and the asymptotic conditional-coverage theorem rests on an explicit consistency assumption plus an external lemma, not on a self-citation chain.

full rationale

I walked the paper's derivation chain. The central object s'(x,y)=F(s(x,y)|x,D) is obtained by fitting a Bayesian predictive CDF on Dcal,1 (Algorithm 1, Step II.2), and the conformal threshold is computed from s' evaluated on a disjoint subset Dcal,2 (Algorithm 1, Step II.3-4). Marginal coverage (Theorem 1) thus follows from standard split-conformal exchangeability (cited to Lei et al. 2018), not from the Bayesian fit — the target coverage is not used as a fitting objective. The asymptotic conditional-coverage claim (Theorem 2) requires Assumption 1, which is stated explicitly as uniform convergence in probability of F(s|x,D) to F(s|x,theta*), with theta* described as the true parameter. The proof shows the empirical quantile of s' is close to the empirical quantile of s''=F(s|x,theta*), then invokes Dheur et al. [2025, Lemma 2] for convergence of the latter quantile to 1-alpha; the uniformity of s'' given X is the standard probability-integral-transform fact following from the assumption. This is a conditional derivation with an identifiable, non-tautological hypothesis; the conclusion is not identical to the assumption, and the Bayesian component is not fitted to coverage. I found no load-bearing self-citation: the authors' own prior work (Cabezas et al. 2024, 2025; Izbicki et al. 2020, 2022) appears in related-work comparisons and as possible extensions, not as justification for Theorem 1 or Theorem 2. The proof's reliance on Dheur et al.'s Lemma 2 is an external result that is not reproduced in the paper, but it is not a self-citation and is not used to forbid alternatives. The main caveat is substantive, not circular: Assumption 1 is strong and is admitted by the authors to be strong, and the MC-dropout/MDN and heteroscedastic BART instantiations are not shown to satisfy it; the paper offers only qualitative empirical robustness. That is a correctness/verification limitation, not a circularity of the derivation. Overall, the paper is self-contained against external benchmarks and its theoretical claim is conditional on a clearly stated assumption, so no circular step reaches the threshold required by the review rules.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The method adds a Bayesian model of the conformal score distribution; the listed hyperparameters are hand-chosen and affect behavior, and the theorems rely on exchangeability plus a uniform-convergence assumption on the Bayesian predictive CDF. No new entities are introduced.

free parameters (3)
  • BART prior depth parameter beta = 0.9 (diffuse) or 0.1 (concentrated) in Appendix D.5; unspecified for main experiments
    Chosen by hand; Appendix D.5 shows it materially changes interval width and AISL, so the method's adaptivity depends on this prior choice.
  • MDN mixture components K = 3
    Fixed by hand; the predictive CDF of scores depends on this capacity choice.
  • GP inducing points = 15, 50, or 150 depending on n
    Set by data size; affects the fidelity of the variational posterior predictive.
assumptions (4)
  • domain assumption Exchangeability of the data (i.i.d. split into training, Dcal1, Dcal2, test)
    Standard conformal prediction assumption; invoked in Theorem 1 and Theorem 2.
  • domain assumption Assumption 1: uniform convergence in probability of the Bayesian predictive CDF F(s|x,D) to the true conditional CDF F(s|x,theta*)
    Load-bearing condition for Theorem 2; requires the Bayesian model to be well-specified and the posterior to concentrate uniformly, which is strong and typically unverifiable.
  • domain assumption The conformal score s has a continuous joint distribution for the upper coverage bound in Theorem 1
    Stated in Theorem 1; used only for the upper bound.
  • standard math Dheur et al. (2025, Lemma 2): the empirical quantile of the true CDF-transform score tends to 1-alpha
    Used in the proof of Theorem 2; treated as an established background result.

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Cite this review

Pith. "Pith review of Epistemic Uncertainty in Conformal Scores: A Unified Approach." pith.science (2026). https://pith.science/paper/ZL2RFNN6

@misc{pith2026250206995,
  author       = {Pith},
  title        = {Pith review of: Epistemic Uncertainty in Conformal Scores: A Unified Approach},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZL2RFNN6}},
  note         = {Machine review of arXiv:2502.06995}
}
abstract

Conformal prediction methods create prediction bands with distribution-free guarantees but do not explicitly capture epistemic uncertainty, which can lead to overconfident predictions in data-sparse regions. Although recent conformal scores have been developed to address this limitation, they are typically designed for specific tasks, such as regression or quantile regression. Moreover, they rely on particular modeling choices for epistemic uncertainty, restricting their applicability. We introduce $\texttt{EPICSCORE}$, a model-agnostic approach that enhances any conformal score by explicitly integrating epistemic uncertainty. Leveraging Bayesian techniques such as Gaussian Processes, Monte Carlo Dropout, or Bayesian Additive Regression Trees, $\texttt{EPICSCORE}$ adaptively expands predictive intervals in regions with limited data while maintaining compact intervals where data is abundant. As with any conformal method, it preserves finite-sample marginal coverage. Additionally, it also achieves asymptotic conditional coverage. Experiments demonstrate its good performance compared to existing methods. Designed for compatibility with any Bayesian model, but equipped with distribution-free guarantees, $\texttt{EPICSCORE}$ provides a general-purpose framework for uncertainty quantification in prediction problems.

Figures

Figures reproduced from arXiv: 2502.06995 by the authors.

Figure 1
Figure 1. A comparison of predictive intervals from standard [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Prediction sets from Adaptive Prediction Sets (APS) versus the proposed [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. EPICSCORE schematic illustration: Given a fitted base model (first panel), we begin by creating a nonconformity score and evaluating it over the calibration set (second panel). We then model the predictive distribution of the conformal score s(X, Y ) using a specified family of models, integrating the epistemic uncertainty about the data-generating process. The predictive CDF of each original score defines a new con… view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: Additional outlier image prediction sets examples. [PITH_FULL_IMAGE:figures/full_fig_p016_4.png]
Figure 5
Figure 5. Figure 5: Additional inlier prediction sets examples. [PITH_FULL_IMAGE:figures/full_fig_p017_5.png]
Figure 6
Figure 6. Figure 6: Left: Prediction set sizes for the top 150 most outlying observations. Right: Prediction set sizes for the top 150 [PITH_FULL_IMAGE:figures/full_fig_p018_6.png]
Figure 7
Figure 7. Figure 7: Comparison of prediction regions produced by [PITH_FULL_IMAGE:figures/full_fig_p024_7.png]
Figure 8
Figure 8. Figure 8: Variation of the Average Interval Score Loss (AISL) as a function of the β hyperparameter in the BART model used within the EPICSCORE framework. Lower AISL values indicate more favorable balance, suggesting that tuning β can significantly affect performance by controll…
Figure 9
Figure 9. Figure 9: Computational cost of EPICSCORE as a function of sample size n (log scale), using the three Bayesian predictive models: BART, MDN and Variational GP. While all three models show comparable running times at moderate scales(e.g., n = 10000), MDNs exhibit significantly be…
Figure 10
Figure 10. Figure 10: Comparison of prediction regions generated by the HPD (Highest Predictive Density) split [Izbicki et al., 2022] [PITH_FULL_IMAGE:figures/full_fig_p027_10.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

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  1. Optimal Conformal Prediction under Epistemic Uncertainty

    stat.ML 2025-05 conditional novelty 6.0 of 10

    Bernoulli Prediction Sets provably produce smallest conformal prediction sets with conditional coverage when the model outputs valid credal sets, generalizing APS.

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Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.