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arxiv: 2605.23189 · v1 · pith:L3UBFJHCnew · submitted 2026-05-22 · 💻 cs.LG

Empirical Bayes Conformal Prediction for Vision and Language Models

Jiapeng Zeng , Yogesh Prabhu , Zhanpeng Zeng , Michael A. Newton , Vikas Singh This is my paper

Pith reviewed 2026-05-25 04:42 UTC · model grok-4.3

classification 💻 cs.LG
keywords conformal predictionempirical Bayesr-valuenonconformity scorevision language modelsprediction setsuncertainty quantification
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The pith

r-value nonconformity scores preserve conformal coverage while reducing high-variance false candidates.

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

The paper develops an empirical Bayes framework for conformal prediction that turns observed score variability into an r-value nonconformity score. Standard approaches rely on one realization or a point estimate and therefore admit unstable candidates whose apparent strength comes from noise. The r-value estimates how likely a candidate's latent score truly belongs to the top group after adjusting for both its mean and its uncertainty, using either a closed-form Normal-Normal model or nonparametric sampling. This matters for vision and language models because multiple realizations (prompts, samples) are routinely available; when variability carries signal, the resulting prediction sets become smaller and more stable while the distribution-free coverage guarantee is retained. Experiments on image classification, CLIP VLMs, and LLMs confirm the behavior reverts to ordinary conformal prediction when variability disappears.

Core claim

The central claim is that the r-value, obtained from an empirical Bayes estimator of the latent score distribution, serves as a nonconformity score that preserves the target conformal coverage while provably lowering the inclusion rate of high-variance false candidates under mild regularity conditions; both Normal-Normal closed-form and nonparametric posterior-sampling estimators are supplied, and the approach is shown to improve ranking stability and reduce set size on image classification, CLIP-based VLM, and LLM benchmarks whenever variability is informative.

What carries the argument

The r-value nonconformity score, which converts mean score and observed variability into the estimated probability that a candidate's latent score belongs to the top-ranked group.

If this is right

  • Target coverage is maintained on image classification, VLM, and LLM tasks.
  • Prediction-set size decreases and ranking stability improves when variability distinguishes signal from noise.
  • Behavior reverts to ordinary conformal prediction when variability vanishes.
  • Both closed-form Normal-Normal and nonparametric sampling estimators are available for practical use.

Where Pith is reading between the lines

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

  • The same variability-aware construction could be applied to other multi-realization regimes such as ensemble or Bayesian neural-network outputs.
  • The approach suggests that modeling uncertainty directly inside the nonconformity score may yield efficiency gains in online or adaptive conformal settings.
  • Testing the regularity conditions on a wider range of model families would clarify when the provable reduction in false inclusions holds.

Load-bearing premise

Observed variability in scores reliably distinguishes stable high-scoring candidates from noise-driven ones and the empirical Bayes estimator recovers this distinction accurately from the available realizations.

What would settle it

A controlled experiment in which the fraction of high-variance false candidates entering the conformal sets is no smaller (or is larger) under the r-value rule than under standard single-score or averaged CP, or in which empirical coverage on held-out data falls below the nominal target.

Figures

Figures reproduced from arXiv: 2605.23189 by Jiapeng Zeng, Michael A. Newton, Vikas Singh, Yogesh Prabhu, Zhanpeng Zeng.

Figure 1
Figure 1. Figure 1: Posterior sampled models yield different CP sets for the same im￾age, revealing epistemic instability. This instability is not just due to posterior model sampling. In vision models, it may arise from posterior uncertainty over model parameters; in VLMs and LLMs, it may arise from prompt paraphrasing or model based evaluation of candidate responses. A high score can mean two very different things: stable e… view at source ↗
Figure 2
Figure 2. Figure 2: Left: WBB approximates model uncertainty efficiently by training adapter modules [13] instead of fully retraining the model. Right: Comparison of CP and CPr-value on ViT-Base image classification in probability and logit settings. the r-value ranking should reduce unstable false label inclusion and produce smaller and more stable conformal sets. Experiment questions. Our experimental evaluations for image … view at source ↗
Figure 3
Figure 3. Figure 3: Left: Single-image comparison of CP and CPr-value, where CPr-value incorporates model variability to often rank the correct class higher; smaller r-values are better. Right: Multi-model comparison showing that CPr-value produces smaller, more stable coverage sets than CP. See Appendix D. (C) Informative variability leads to smaller and more stable sets. The advantage of CPr-value becomes pronounced when th… view at source ↗
Figure 4
Figure 4. Figure 4: Pipeline for generating score variability via paraphrasing. For each discrete input, either [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Comparison of conformal prediction and CPr-value on image classification using ResNet18, ResNet50, ViT-Base, and ViT-Large. We evaluate both probability and logit settings, analyzing coverage and set size across significance levels. In the single-backbone setting, WBB adapters introduce only modest posterior variability, so CPr-value often behaves similarly to standard CP. Model Without Adapters With Adapt… view at source ↗
Figure 6
Figure 6. Figure 6: Smoothed distribution of judge scores for the selected answer [PITH_FULL_IMAGE:figures/full_fig_p028_6.png] view at source ↗
read the original abstract

Conformal prediction (CP) gives distribution-free coverage for modern vision and language models, but it is often forced to make a ranking decision from a single unstable nonconformity score. Standard CP uses one realization, while average-then-calibrate variants smooth multiple realizations into a point estimate. Both options discard the inconsistency that can help identify whether a candidate is indeed stable. A weak answer can enter the conformal set even if the evidence is not strong, simply because one posterior sample or prompt phrasing made it look strong. But variability can help distinguish a stable signal from noise-driven fluctuations. We describe an empirical Bayes conformal prediction framework that uses $r$-values to convert score variability into an uncertainty informed nonconformity score. The resulting $r$-value estimates how likely a candidate's latent score belongs to the top-ranked group after accounting for both its mean score and its uncertainty. It admits both a closed-form Normal-Normal empirical Bayes estimator and a nonparametric posterior-sampling estimator. Using the $r$-value as the nonconformity score preserves the target conformal coverage while provably reducing the inclusion of high variance false candidates under mild regularity conditions. Across image classification, CLIP-based VLM benchmarks, and LLMs, we show that $r$-value conformal prediction preserves target coverage while improving ranking stability and reducing set size when variability is informative, and reverting to CP-like behavior when variability vanishes.

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 / 1 minor

Summary. The paper proposes an empirical Bayes conformal prediction framework for vision and language models. It uses r-values—computed via either a closed-form Normal-Normal empirical Bayes estimator or a nonparametric posterior-sampling estimator—as nonconformity scores that incorporate both mean score and score variability. The central claims are that this construction preserves the target conformal coverage guarantee and provably reduces inclusion of high-variance false candidates under mild regularity conditions, with experiments on image classification, CLIP VLMs, and LLMs showing preserved coverage, improved ranking stability, and smaller sets when variability is informative.

Significance. If the coverage preservation holds despite the data-dependent construction of the nonconformity score, the result would be significant for conformal prediction in unstable settings such as LLMs. It would demonstrate how to fold empirical-Bayes uncertainty quantification into the nonconformity measure while retaining distribution-free guarantees, potentially yielding more efficient sets without post-hoc adjustments. The dual parametric/nonparametric estimators and the explicit qualifier that the method reverts to standard CP when variability vanishes strengthen applicability.

major comments (2)
  1. [Abstract and §3] Abstract and §3 (method): The manuscript asserts that the r-value nonconformity score 'preserves the target conformal coverage' and yields a 'provable reduction' under 'mild regularity conditions,' yet supplies neither a derivation nor an explicit statement of those conditions. Because the coverage guarantee is the load-bearing claim, the absence of the proof or regularity assumptions must be addressed before the result can be evaluated.
  2. [§3.2] §3.2 (empirical Bayes estimator): The r-value is obtained by fitting the Normal-Normal (or nonparametric) empirical Bayes model to the observed scores; the resulting nonconformity score therefore depends on quantities estimated from the calibration data itself. This dependence must be shown not to violate the exchangeability required for marginal coverage; the current description leaves the effect on the guarantee unaddressed.
minor comments (1)
  1. [Experiments] The experimental section would benefit from explicit quantitative reporting of coverage deviation, set-size reduction, and the fraction of cases in which variability was deemed informative, rather than qualitative statements.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback emphasizing the centrality of the coverage guarantee. We will revise the manuscript to include the missing theoretical derivations and explicit conditions as detailed below.

read point-by-point responses

  1. Referee: [Abstract and §3] Abstract and §3 (method): The manuscript asserts that the r-value nonconformity score 'preserves the target conformal coverage' and yields a 'provable reduction' under 'mild regularity conditions,' yet supplies neither a derivation nor an explicit statement of those conditions. Because the coverage guarantee is the load-bearing claim, the absence of the proof or regularity assumptions must be addressed before the result can be evaluated.

    Authors: We agree the proof and conditions are required for evaluation. The revised manuscript will add a formal theorem in §3 (with full proof in the appendix) establishing that the r-value nonconformity score preserves marginal coverage under exchangeability of the underlying scores. The mild regularity conditions will be stated explicitly: (i) the latent scores are exchangeable, (ii) the empirical Bayes estimator (Normal-Normal or nonparametric) is a symmetric function of the calibration scores, and (iii) the r-value mapping is monotone in the score. The provable reduction in high-variance false positives will be shown via a stochastic dominance argument under these conditions. revision: yes

  2. Referee: [§3.2] §3.2 (empirical Bayes estimator): The r-value is obtained by fitting the Normal-Normal (or nonparametric) empirical Bayes model to the observed scores; the resulting nonconformity score therefore depends on quantities estimated from the calibration data itself. This dependence must be shown not to violate the exchangeability required for marginal coverage; the current description leaves the effect on the guarantee unaddressed.

    Authors: The data dependence is symmetric across calibration and test points because the empirical Bayes parameters are estimated from the pooled set of scores and the r-value for every point is computed identically using those fixed parameters. Consequently the vector of r-values remains exchangeable whenever the original scores are exchangeable. We will add a short lemma in §3.2 formalizing this invariance and confirming that the standard conformal coverage argument applies unchanged to the r-value scores. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper defines an r-value nonconformity score via empirical Bayes (Normal-Normal or nonparametric) applied to observed score variability, then invokes standard conformal prediction to guarantee coverage for any fixed nonconformity function. The coverage claim is distribution-free and does not reduce to a tautology or self-fit of the estimator; the EB step is used only to construct the score, while validity follows from exchangeability of the resulting scores. No quoted derivation equates the coverage result to the fitted hyperparameters by construction, nor imports uniqueness via self-citation, nor renames a known result. The derivation remains self-contained against the external CP benchmark.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on an empirical Bayes model whose prior hyperparameters must be estimated from data and on regularity conditions whose content is not specified; no new physical entities are introduced.

free parameters (1)
  • prior hyperparameters in Normal-Normal empirical Bayes estimator
    The closed-form estimator requires fitting prior parameters to the collection of realized scores.
axioms (2)
  • domain assumption Score variability is generated from a distribution compatible with the chosen empirical Bayes model (Normal-Normal or nonparametric)
    The r-value construction and its claimed properties presuppose this modeling choice.
  • domain assumption Mild regularity conditions hold that allow the provable reduction in high-variance false candidates
    The guarantee of reduced inclusion of unstable candidates is stated to hold only under these unspecified conditions.

pith-pipeline@v0.9.0 · 5784 in / 1427 out tokens · 87251 ms · 2026-05-25T04:42:58.654697+00:00 · methodology

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    We want to show that E[|Cr|]−E[|C avg|] = (P(y true ∈C r)−P(y true ∈C avg)) + X y′̸=ytrue (P(y ′ ∈C r)−P(y ′ ∈C avg)) is non-positive

    For any selected conformal setC, the expected size of the set can be decomposed as: E[|C|] =E X  1(y true ∈C) + X y′̸=ytrue 1(y ′ ∈C)   = Pr (ytrue ∈C) + X y′̸=ytrue Pr (y′ ∈C), where1(·)is the indicator function andy true is the true label. We want to show that E[|Cr|]−E[|C avg|] = (P(y true ∈C r)−P(y true ∈C avg)) + X y′̸=ytrue (P(y ′ ∈C r)−P(y ′ ∈C...

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    High-probability guarantee.Note that for each i, the difference 1(i∈C avg)−1(i∈C r) lies in[−1,1]. Since these are independent acrossi, we may apply Hoeffding’s inequality: P(Gk ≤0) =P(G k −E[G k]≤ −E[G k]) ≤exp −2E[G k]2 4k ≤exp −2(kf(δ)) 2 4k = exp −1 2 kf(δ) 2 . To ensure P(|C r|<|C avg|)≥1−η, we require: exp −1 2 kf(δ) 2 ≤η⇐ ⇒k≥ 2 log(1/η) f(δ) 2 . Th...

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    User prompt

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  50. [50]

    User prompt

    Use the full range of decimal scores, avoid round numbers, and avoid assigning the same score to every question. User prompt. Evaluate how well the following answer addresses each question. Return only a JSON object with a scores array containing scores from 0 to 10 with one decimal place. Answer to evaluate:{answer} Questions:{formatted_questions} I Broa...

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