Conformal Sets in Multiple-Choice Question Answering under Black-Box Settings with Provable Coverage Guarantees

Guang Yang; Xinyang Liu

REVIEW 3 major objections 6 minor 22 references

Conformal Sets in Multiple-Choice Question Answering under Black-Box Settings with Provable Coverage Guarantees

T0 review · 3 major / 6 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read This paper claims that frequency-based predictive entropy from repeated black-box samples matches or beats logit-based entropy for distinguishing correct multiple-choice answers, and that conformal prediction sets built on those scores have

desk verdict Useful scan of a known idea—frequency-based entropy plus conformal prediction—but the proof has a quantile-index bug and the AUROC claim is weaker than advertised. read the letter →

arxiv 2508.05544 v1 pith:PESDJIY7 submitted 2025-08-07 cs.CL cs.AI

Guang Yang , Xinyang Liu This is my paper

classification cs.CLcs.AI

keywords conformalpredictionuncertaintyquantificationmultiple-choicequestionansweringblack-boxLLMpredictiveentropysamplingfrequencycoverageguaranteeAUROC

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 tries to establish that, for multiple-choice question answering with large language models, an uncertainty score computed from the empirical frequency of repeated black-box samples is at least as good as the usual softmax-logit predictive entropy at telling correct from incorrect answers, and that feeding this score into conformal prediction yields prediction sets whose probability of containing the true answer is at least $1-\alpha$. The motivation is practical: APIs often hide logits, so uncertainty must be built from observable outputs. Across six LLMs and four datasets, the frequency-based predictive entropy matched or beat logit-based entropy in AUROC, and the conformal sets kept empirical miscoverage below the user-specified risk level. If the finding holds, sampling frequency becomes a drop-in substitute for logit access in distribution-free uncertainty quantification for black-box LLMs.

What carries the argument

The load-bearing object is frequency-based predictive entropy, $\mathrm{PE}_{\mathrm{freq}}$, built from the empirical answer frequencies of $M$ independent black-box samples, with the modal answer used as the reference prediction. This score replaces the internal logit probability in the conformal nonconformity score $s = 1 - F(x)_y$; the quantile threshold $\hat q$ then produces prediction sets whose coverage follows from the exchangeability and rank identity expressed in the paper's Eqs. (8)-(9). That identity is what converts sampling-based uncertainty into a finite-sample coverage guarantee.

What would settle it

On MMLU, calibrate on a random half of Clinical Knowledge questions and test on Law questions, repeating 100 splits at $\alpha=0.2$ with the paper's $M=20$ and temperature 1.0; if the empirical miscoverage rate is systematically above 0.2, the claimed automatic coverage guarantee fails.

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Extended reading notes

Core claim

The central claim is that predictive entropy computed from the empirical frequency distribution of $M$ repeated samples, with the most frequent answer serving as the reference, captures prediction confidence as well as or better than predictive entropy computed from internal logits. Concretely, the paper defines $\hat P(a) = \frac{1}{M}\sum_{m=1}^M \mathbb{I}(a_m=a)$ and $\mathrm{PE}_{\mathrm{freq}} = -\sum_i \hat P_i(a)\log_b \hat P_i(a)$, then uses these frequency scores in the standard conformal quantile construction: nonconformity scores $s_i = 1 - F(x_i)_{y_i^*}$, threshold $\hat q = \mathrm{quantile}(\{s_i\}_{i=1}^n, \lfloor(n+1)(1-\alpha)\rfloor/n)$, and prediction set $C(x_{n+1}) = \

Load-bearing premise

The coverage guarantee collapses if the calibration questions and the test question are not drawn from the same distribution: any shift in subject mix, sampling temperature, or the number of samples $M$ breaks the rank argument that produces the $1-\alpha$ bound.

Editorial extensions

If this is right

Black-box LLM APIs can quantify uncertainty with no logit access: sample the same question $M$ times and use answer frequencies to score confidence.
Those scores plug into conformal prediction to return a set of options guaranteed, under exchangeability, to contain the correct answer with probability at least $1-\alpha$.
At a fixed risk level, models with lower intrinsic uncertainty yield smaller prediction sets, so average set size can serve as a reliability diagnostic.
Frequency-based uncertainty transfers across model families (Llama, Qwen, Vicuna) and across general and medical MCQA benchmarks without retraining or model-specific calibration.

Reading between the lines

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

The paper evaluates only same-dataset random splits; a natural extension it does not make is to test the guarantee under subject-mix shift, temperature change, or different sampling budget $M$, where the exchangeability premise may fail.
Because the score is a histogram over $M$ samples, it ignores ordering and semantic similarity among options; smoothing frequencies with a small prior could improve AUROC when options are close.
The same modal-frequency construction could be adapted to free-form generation by clustering sampled responses before counting, extending the idea beyond multiple-choice formats; the paper does not test this.
A practical design question left open is how small $M$ can be before frequency-based PE loses its edge over logits; the paper fixes $M=20$ rather than sweeping it.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Quick take: this is a clean, small empirical study of an idea that has been floating around—use sampling frequency instead of logits as the score for conformal prediction—but the paper overstates both halves of its claim. The coverage theorem in Eq. (9) is wrong as written: floor((n+1)(1−α))/(n+1) is not ≥ 1−α. For n=100, α=0.1 you get 0.891 < 0.90. The fix is trivial (use ceiling), but until it's made the "provable coverage" headline isn't supported. Eq. (5) is also inconsistent with the rank argument, and the empirical EMR plots just illustrate the standard conformal rank argument; they can't repair the proof.

The AUROC comparison is the real content. The authors ran six models across four multiple-choice datasets, comparing predictive entropy from empirical frequencies with logit-based entropy. The pattern is mostly positive: 18 of 24 cells favor the frequency score, and average AUROC improves on MMLU and MedMCQA. But "generally outperforms" is doing real work: MedQA's average favors logits, several cells are essentially ties, and there are no error bars or significance tests. The intro singles out the best single cell (Qwen2.5-3B on MedMCQA, +2%), which is cherry-picking. The honest claim is "comparable, sometimes slightly better."

There are also presentation gaps. Eq. (4)'s score F is never specified as logit- or frequency-based, so it's unclear which score generated the EMR plots. The paper doesn't compare prediction-set sizes against a logit-based conformal baseline, only APSS for the frequency sets. And for a paper using sampling-frequency confidence, the absence of self-consistency references is conspicuous—that's the obvious prior work. Also, the hyperparameter section mentions CoQA and TriviaQA, which aren't in the dataset list, suggesting sloppy copy-paste.

What's good: the experiment covers a decent spread of model sizes and domains, and the EMR curves are a reasonable sanity check. The writing is mostly clear. The errors are fixable, which is why I'd send it to review rather than desk reject. The authors need to fix the quantile, state the score explicitly, add error bars, and tone down the abstract.

Referee Report

3 major / 6 minor

Summary. The paper proposes a frequency-based predictive-entropy score for uncertainty quantification in multiple-choice question answering under black-box LLM settings. For each question, the model is sampled M times; the empirical frequency of each answer is used to compute predictive entropy, replacing softmax-logit entropy. These scores are plugged into split conformal prediction to construct label sets with a stated finite-sample coverage guarantee. The authors evaluate AUROC of frequency-based vs. logit-based entropy across six models and four datasets, and report empirical miscoverage rates and average prediction-set sizes. The central claims are that frequency-based entropy generally outperforms logit-based entropy in distinguishing correct from incorrect answers and that the conformal sets provide provable coverage at user-specified levels.

Significance. If the coverage proof is corrected and the AUROC claim appropriately qualified, the paper is a useful applied contribution: it provides a black-box-compatible nonconformity score for MCQA and demonstrates that sampled-answer frequencies are a workable substitute for logits in standard conformal prediction. The AUROC comparison is genuinely independent of the conformal construction and is the most informative part of the empirical study. The theoretical content is not novel—the coverage argument is the standard conformal rank argument—but the paper's value lies in its systematic evaluation across six LLMs and four datasets. Credit is due for repeating calibration/test splits and reporting EMR with means and standard deviations, although the empirical EMR plots are illustrations of the standard theorem rather than independent evidence that the proposed score is better.

major comments (3)

[§3.2, Eq. (9)] The stated coverage guarantee is false as written. Eq. (9) claims P(y*_{n+1} ∈ C(x_{n+1})) = floor((n+1)(1−α))/(n+1) ≥ 1−α. For n=100, α=0.10 this gives 90/101 ≈ 0.891 < 0.90. For exchangeable scores, a set built from the k-th order statistic has coverage k/(n+1); to guarantee at least 1−α one must take k = ceil((n+1)(1−α)). Eq. (5) is also inconsistent: it defines the quantile with denominator n and ambiguous |·|/floor notation, whereas Eq. (9) uses a floor index over n+1. This is load-bearing because the headline 'provable coverage guarantee' rests entirely on this step. The fix is local, but the theorem and the statements in Sections 4.3 and 5 must be corrected.
[§4.2, Table 1 and Abstract/Conclusion] The claim that frequency-based PE 'generally outperforms' or 'outperforms' logit-based PE is stronger than the data support. In Table 1, frequency-based AUROC is higher in 16 of 24 model-dataset cells and lower in 8; the MedQA average favors logits (0.5986 vs. 0.5975), and several differences are tiny (e.g., MMLU-Pro Llama-3.2-1B: 0.5293 vs. 0.5285). No error bars, confidence intervals, or significance tests are reported. The abstract and conclusion should be tempered to 'comparable, with gains on several datasets,' or the authors should add paired statistical tests to justify 'generally outperforms.'
[§3.2, Eq. (4)] The nonconformity score is under-specified. Eq. (4) defines s_i = 1 − F(x_i)_{y*_i}, but F was introduced in §3.1 as a general classification model. It is never stated whether F is the logit-based score or the frequency-based estimator of Eq. (1). If F is logit-based, the method is not black-box; if F is frequency-based, the paper must define exactly how F(x)_y is computed for each candidate label and verify that the resulting scores are exchangeable. This ambiguity also determines which of the EMR curves (Figs. 1–3) validate the proposed frequency-based method.

minor comments (6)

[§3.2, Eq. (2)] The predictive-entropy formula is mis-indexed. It should sum over candidate answers a ∈ A, not over i = 1,...,n (which is used for calibration samples). As written, the expression −∑_{i=1}^n P_i(a) log_b P_i(a) is dimensionally inconsistent.
[§3.2, Eq. (8)] Eq. (8) writes P(s_{n+1} ≤ s_i) = i/(n+1), but s_i was defined as a raw calibration score in Eq. (4). The rank argument requires order statistics; use s_{(i)} and state that this holds for the i-th smallest calibration score.
[§1] The Introduction says 'six datasets and four models'; the paper evaluates six models and four datasets. Please correct.
[§4.1] The hyperparameter paragraph mentions maximum generation lengths for CoQA and TriviaQA, but these datasets are not used in the evaluation. Either remove the mention or add the datasets.
[Figs. 1–3 and §4.3] The EMR plots are empirical checks of the standard conformal theorem for the proposed score; they do not, by themselves, demonstrate that frequency-based scores are superior to logit-based scores. The text should frame them as sanity checks, not as independent validation of the substitution claim.
[§3.2] The coverage guarantee requires exchangeability of the calibration and test scores. The paper should state explicitly that a change of subject mix, sampling temperature, or M between calibration and test would void the guarantee, since the experiments only cover the i.i.d. split setting.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the AUROC comparison is independent and the conformal coverage argument is the standard rank-based theorem, not a self-referential construction.

full rationale

After walking the derivation chain, no circular step is present. The frequency-based predictive entropy in Eqs. (1)-(2) is an independently defined score computed from repeated sampling, and the claim that it distinguishes correct from incorrect answers is evaluated by AUROC against reference labels; no parameter is fit to those labels, and the score is not defined in terms of the conformal coverage outcome it later feeds. The conformal coverage guarantee in Eqs. (8)-(9) is the standard exchangeability/rank argument taken from the cited conformal prediction literature, not from the authors' own prior work, and the empirical miscoverage plots are a sanity check of that theorem rather than the evidence from which the theorem is derived. There are no self-citations, no fitted inputs renamed as predictions, and no uniqueness or ansatz imported from the authors' prior papers. A possible mathematical error in Eq. (9) regarding floor vs. ceiling quantile indices would be a correctness concern, not a circularity concern, and does not affect this verdict.

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

The method is standard conformal machinery applied to a hand-chosen frequency score. The coverage guarantee is fully inherited from the cited CP framework; the only genuinely new degrees of freedom are the sampling hyperparameters (M, temperature, split ratio), which are chosen by hand with no sensitivity analysis, and the implicit modeling assumption that empirical frequencies faithfully encode the LLM's predictive distribution. No new entities are postulated.

free parameters (3)

M (number of candidate samples per prompt) = 20
Chosen by hand with no sensitivity analysis; the entropy estimate and the AUROC comparison depend on it (Sec. 4.1).
sampling temperature = 1.0
Set to 1.0 to ensure diverse outputs; no other temperatures tested (Sec. 4.1).
calibration/test split ratio = 0.5
By default 50/50; coverage and set-size results depend on calibration size (Sec. 4.1).

assumptions (4)

domain assumption Calibration questions and the test question are i.i.d. (exchangeable nonconformity scores)
Eq. (8) invokes i.i.d. to justify P(s_{n+1} <= s_i) = i/(n+1); the coverage guarantee is conditional on this.
standard math Split conformal quantile theorem with rank-based coverage
Eq. (9) relies on the standard conformal rank argument; note the floor in Eq. (9) makes the written inequality false in general, the ceiling is required.
domain assumption The empirical frequency distribution over M samples approximates the LLM's output distribution
Implicit in treating Phat(a) as a substitute for softmax probabilities (Eqs. 1-2, used throughout Sec. 4).
domain assumption Sampled answers are independent draws conditional on the prompt
Eq. (1) treats the M samples as i.i.d. draws; order effects, caching, or generation noise would bias the frequencies.

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Cite this review

Pith. "Pith review of Conformal Sets in Multiple-Choice Question Answering under Black-Box Settings with Provable Coverage Guarantees." pith.science (2026). https://pith.science/paper/PESDJIY7

@misc{pith2026250805544,
  author       = {Pith},
  title        = {Pith review of: Conformal Sets in Multiple-Choice Question Answering under Black-Box Settings with Provable Coverage Guarantees},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PESDJIY7}},
  note         = {Machine review of arXiv:2508.05544}
}

read the original abstract

Large Language Models (LLMs) have shown remarkable progress in multiple-choice question answering (MCQA), but their inherent unreliability, such as hallucination and overconfidence, limits their application in high-risk domains. To address this, we propose a frequency-based uncertainty quantification method under black-box settings, leveraging conformal prediction (CP) to ensure provable coverage guarantees. Our approach involves multiple independent samplings of the model's output distribution for each input, with the most frequent sample serving as a reference to calculate predictive entropy (PE). Experimental evaluations across six LLMs and four datasets (MedMCQA, MedQA, MMLU, MMLU-Pro) demonstrate that frequency-based PE outperforms logit-based PE in distinguishing between correct and incorrect predictions, as measured by AUROC. Furthermore, the method effectively controls the empirical miscoverage rate under user-specified risk levels, validating that sampling frequency can serve as a viable substitute for logit-based probabilities in black-box scenarios. This work provides a distribution-free model-agnostic framework for reliable uncertainty quantification in MCQA with guaranteed coverage, enhancing the trustworthiness of LLMs in practical applications.

Figures

Figures reproduced from arXiv: 2508.05544 by the authors.

**Figure 1.** Empirical Miscoverage Rate on MMLU 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Alpha 0.0 0.2 0.4 0.6 0.8 Empirical Miscoverage Rate (EMR) diagonal vicuna-7b-v1.5 Qwen2.5-3B-Instruct Llama-3.2-1B Qwen2.5-7B-Instruct Llama-3.1-8B-Instruct vicuna-13b-v1.5 (a) Health 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Alpha 0.0 0.2 0.4 0.6 0.8 Empirical Miscoverage Rate (EMR) diagonal vicuna-7b-v1.5 Qwen2.5-3B-Instruct Llama-3.2-1B Qwen2.5-7B-… view at source ↗

**Figure 2.** Empirical Miscoverage Rate on MMLU Pro rate (EMR) -defined as the proportion of test samples where the prediction set fails to include the correct answer - to ensure it adheres to user-specified thresholds. Additionally, we report the average prediction set size (APSS) as a metric of efficiency, where smaller sizes denote enhanced practical utility. Hyperparameters. We set the maximum length of generation based on d… view at source ↗

**Figure 3.** Empirical Miscoverage Rate 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Alpha 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Average Prediction Set Size (APSS) vicuna-7b-v1.5 Qwen2.5-3B-Instruct Llama-3.2-1B Qwen2.5-7B-Instruct Llama-3.1-8B-Instruct vicuna-13b-v1.5 (a) MEDMCQA 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Alpha 0 1 2 3 4 5 Average Prediction Set Size (APSS) vicuna-7b-v1.5 Qwen2.5-3B-Instruct Llama-3.2-1B Qwen2.5-7B-Instruct Llam… view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Average Prediction Set Size sampled outputs—achieves AUROC comparable to logit-based methods, thus validating sampling frequency as a viable logit substitute for UQ in black-box LLMs. Experiments show that frequency-based methods yield AUROC values essentially equivale…

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Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

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arXiv 2021

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Anastasios Nikolas Angelopoulos, Stephen Bates, Adam Fisch, Lihua Lei, and Tal Schuster. Conformal risk control. In The Twelfth International Conference on Learning Representations, 2024

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Jinhe Bi, Yifan Wang, Danqi Yan, Xun Xiao, Artur Hecker, V olker Tresp, and Yunpu Ma. Prism: Self-pruning intrinsic selection method for training-free multimodal data selection. arXiv preprint arXiv:2502.12119, 2025

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Jinhe Bi, Danqi Yan, Yifan Wang, Wenke Huang, Haokun Chen, Guancheng Wan, Mang Ye, Xun Xiao, Hinrich Schuetze, V olker Tresp, et al. Cot-kinetics: A theoretical modeling assessing lrm reasoning process. arXiv preprint arXiv:2505.13408, 2025

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