arxiv: 2605.12398 · v1 · submitted 2026-05-12 · 💻 cs.CL · cs.IR

Recognition: 2 theorem links

· Lean Theorem

Question Difficulty Estimation for Large Language Models via Answer Plausibility Scoring

Adam Jatowt, Bhawna Piryani, Jamshid Mozafari

Pith reviewed 2026-05-13 05:24 UTC · model grok-4.3

classification 💻 cs.CL cs.IR

keywords question difficulty estimationanswer plausibility scoreslarge language modelsquestion answeringentropyQA evaluationhuman alignment

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The pith

Entropy of plausibility scores over candidate answers estimates question difficulty for large language models.

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

The paper proposes Q-DAPS as a way to gauge how hard a question is for LLMs by measuring the entropy in the plausibility scores the model assigns to different possible answers. This replaces reliance on readability formulas, retrieval signals, or popularity counts that may miss the actual reasoning demands modern models face. Experiments across TriviaQA, Natural Questions, MuSiQue, and QASC show the approach beats standard baselines while remaining stable under changes in hyperparameters, question styles, model sizes, and plausibility scoring methods. Human raters' difficulty judgments line up closely with the scores Q-DAPS produces. The result is a practical, scalable signal for evaluating and improving question-answering performance.

Core claim

Q-DAPS estimates difficulty by computing the entropy of plausibility scores assigned to a set of candidate answers for each question. Greater entropy signals higher difficulty because the model treats multiple answers as comparably plausible, indicating deeper reasoning challenges. This measure outperforms prior baselines on four QA datasets, holds steady across hyperparameter and model variations, and matches human assessments of difficulty.

What carries the argument

Q-DAPS, which derives a difficulty score directly from the entropy of the distribution of plausibility scores over candidate answers.

If this is right

Q-DAPS outperforms existing baselines on TriviaQA, NQ, MuSiQue, and QASC.
The method remains robust across hyperparameter changes, question types, model sizes, and plausibility estimation approaches.
Q-DAPS difficulty estimates align closely with human judgments.
Ablation studies confirm reliability in realistic evaluation settings.

Where Pith is reading between the lines

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

The approach could support curriculum construction by ranking questions according to model-specific difficulty.
It offers a route to diagnose which reasoning patterns cause models to assign similar plausibility to several answers.
Scalability allows difficulty labeling of very large question collections without additional human annotation.

Load-bearing premise

The entropy of plausibility scores over candidate answers captures the reasoning challenges posed to LLMs rather than merely reflecting surface-level answer distributions or calibration artifacts.

What would settle it

A large-scale test in which Q-DAPS scores fail to predict actual LLM accuracy rates on held-out questions or diverge from human difficulty ratings on a new dataset would falsify the central claim.

Figures

Figures reproduced from arXiv: 2605.12398 by Adam Jatowt, Bhawna Piryani, Jamshid Mozafari.

**Figure 1.** Figure 1: Two examples from the NQ (green) and TriviaQA (blue) datasets are presented, each showing the correct answer, 10 candidate answers (selected from 20 generated candidates), their normalized plausibility scores, and the computed difficulty score. The green colored example illustrates low entropy (an easier question), while the blue colored example demonstrates high entropy (a harder question). medium, or h… view at source ↗

**Figure 2.** Figure 2: The Q-DAPS method comprises three stages: Candidate Generation, which produces candidate answers and their plausibility scores; Popularity Debiasing, which adjusts these plausibility scores based on candidate popularity; and Scoring, which computes the final difficulty score for the given question. designed for humans or retrieval statistics tailored to IR systems, and (2) it directly links difficulty esti… view at source ↗

**Figure 3.** Figure 3: Prompt used for listwise candidate answer [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

**Figure 4.** Figure 4: Kendall’s τ (left chart) and Spearman’s ρ (right chart) correlation coefficients across datasets comparing candidate answer ordering with Popularity, Plausibility scores, and their interplay. 3 Experimental Setup 3.1 Datasets We use TriviaQA (Joshi et al., 2017) and Natural Questions (NQ) (Kwiatkowski et al., 2019) datasets for simple questions, as well as MuSiQue (Trivedi et al., 2022) and QASC (Khot et… view at source ↗

**Figure 6.** Figure 6: The template of the Excel sheet of questions [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗

**Figure 7.** Figure 7: The template of the Excel sheet of questions [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗

**Figure 5.** Figure 5: Results of human evaluation. The blue columns indicate each evaluator’s accuracy on easy questions, while the orange columns represent their accuracy on hard questions. quiring only a single prompt with complexity O(1), compared to O(n) for Pointwise and O(n 2 ) for Pairwise, as reported in [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗

**Figure 8.** Figure 8: The placeholder <question> represents the question, <ground_truth> indicates the correct answer, and <candidate> shows the answer generated by different LLMs. E Metrics Details In this section, we provide detailed descriptions of the evaluation metrics used throughout the paper, including both GPT-Eval and Cohen’s d. E.1 GPT-Eval In this section, we describe the motivation of using GPT-Eval instead of tok… view at source ↗

**Figure 9.** Figure 9: This prompt uses five-shot examples to guide [PITH_FULL_IMAGE:figures/full_fig_p019_9.png] view at source ↗

**Figure 10.** Figure 10: The placeholder <question> represents the given question, while <ground_truth> denotes its correct answer. Candidate answer is represented by <candidate_answer>, along with a plausibility score (<plausibility_score>) and a justification (<justification>) explaining both the answer choice and the reasoning behind its assigned score. F Additional Experiments In this section, we present additional experiment… view at source ↗

**Figure 11.** Figure 11: An example from TriviaQA of the pointwise scenario showing that most plausibility scores for the [PITH_FULL_IMAGE:figures/full_fig_p021_11.png] view at source ↗

**Figure 12.** Figure 12: The placeholder <question> represents the given question, while <ground_truth> denotes its correct answer. Two candidate answers, <candidate_1> and <candidate_2>, are compared, and the LLM is asked to select which one is more likely to be correct, along with a justification. sign a score of one to most candidates. In contrast, the listwise scenario produces a more diverse range of plausibility scores. Ad… view at source ↗

**Figure 14.** Figure 14: α robustness across different values from 0 to 1 on various question categories, including Simple, Complex, and Entire question types. Question Type Optimal α Optimal # of Cans d Simple 0.49 7 0.8299 Complex 0.52 8 0.8423 Entire 0.52 8 0.8142 [PITH_FULL_IMAGE:figures/full_fig_p022_14.png] view at source ↗

**Figure 15.** Figure 15: An example from the TriviaQA dataset illustrating how including the correct answer among the candidate [PITH_FULL_IMAGE:figures/full_fig_p023_15.png] view at source ↗

**Figure 16.** Figure 16: The placeholder <question> represents the given question. Each candidate answer is denoted by <candidate_answer>, accompanied by an initial plausibility score (<plausibility_score>) and a justification (<justification>) explaining both the answer choice and the rationale behind its plausibility score. G Gold Answer Inclusion In this section, we show the impact of including the correct answer among candi… view at source ↗

read the original abstract

Estimating question difficulty is a critical component in evaluating and improving large language models (LLMs) for question answering (QA). Existing approaches often rely on readability formulas, retrieval-based signals, or popularity statistics, which may not fully capture the reasoning challenges posed to modern LLMs. In this paper, we introduce Q-DAPS (Question Difficulty based on Answer Plausibility Scores) method, a novel approach that estimates question difficulty by computing the entropy of plausibility scores over candidate answers. We systematically evaluate Q-DAPS across four prominent QA datasets-TriviaQA, NQ, MuSiQue, and QASC-demonstrating that it consistently outperforms baselines. Moreover, Q-DAPS shows strong robustness across hyperparameter variations and question types. Extensive ablation studies further show that Q-DAPS remains robust across different plausibility estimation paradigms, model sizes, and realistic settings. Human evaluations further confirm strong alignment between Q-DAPS's difficulty estimates and human judgments of question difficulty. Overall, Q-DAPS provides an interpretable, scalable, and bias-resilient approach to question difficulty estimation in modern QA systems.

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.

Desk Editor's Note private letter to a colleague

Q-DAPS uses entropy over plausibility scores to rank question difficulty and beats some baselines on four datasets while tracking human judgments, but the signal may track calibration or answer frequency more than reasoning load.

read the letter

The core claim is that computing entropy across an LLM's plausibility scores for candidate answers gives a better difficulty estimate than readability, retrieval, or popularity baselines. They test this on TriviaQA, Natural Questions, MuSiQue, and QASC, run ablations over model sizes and plausibility methods, check robustness to hyperparameters and question types, and add a human study showing alignment with difficulty ratings. That package of experiments is the main concrete contribution and it is executed at a reasonable scale for this kind of work. The method itself is simple to reproduce once the candidate sets are fixed, which helps practical adoption for benchmark curation. The human evaluation is a clear positive; it moves the metric beyond purely automatic comparisons. The soft spot sits in the interpretation step. Entropy of plausibility scores is produced by the same class of model that will later answer the questions, so it can easily reflect how sharply the model concentrates on frequent training answers or surface cues rather than the number of reasoning steps required. The paper does not appear to include controls that separate those factors, such as holding answer distribution constant while varying reasoning depth or testing on synthetic items where difficulty is known by construction. Without those, the claim that the metric isolates LLM reasoning challenges stays partly assumptive. The robustness results are useful but do not resolve the underlying question of what the entropy is actually measuring. This paper is aimed at people who build or maintain QA evaluation suites and want a lightweight, automatic way to order items by expected difficulty. A reader already working on dataset filtering or adaptive testing will find the implementation details and the four-dataset comparison worth looking at. It is not a foundational rethinking of difficulty, but the empirical package is solid enough to merit referee time. I would send it to review rather than desk reject, with the main request being clearer evidence that the signal tracks reasoning load beyond calibration artifacts.

Referee Report

2 major / 1 minor

Summary. The manuscript introduces Q-DAPS, a method for estimating question difficulty for LLMs in QA by computing the entropy of plausibility scores over candidate answers. It evaluates the approach on four datasets (TriviaQA, NQ, MuSiQue, QASC), claiming consistent outperformance over baselines, robustness across hyperparameters, question types, plausibility paradigms, and model sizes, plus alignment with human difficulty judgments. The method is positioned as more interpretable and bias-resilient than readability formulas, retrieval signals, or popularity statistics.

Significance. If the empirical results hold with proper controls and quantitative detail, Q-DAPS could offer a scalable way to assess LLM-specific reasoning load in QA, aiding dataset curation and model evaluation beyond surface features. The claimed robustness and human alignment would add practical value, but the absence of metrics, error bars, or baseline specifications in the provided abstract makes it impossible to gauge the contribution's strength at present.

major comments (2)

[Abstract] Abstract: the central claims of 'consistent outperformance' and 'strong robustness' are asserted without any quantitative results, error bars, baseline implementation details, or statistical tests. This omission is load-bearing because it prevents verification of whether Q-DAPS actually improves on existing methods or merely reflects post-hoc choices.
[Method] Method (entropy of plausibility scores): the assumption that this entropy isolates reasoning challenges posed to LLMs is not secured against confounds such as model calibration artifacts, training-data answer frequency, or lexical cues in the candidate sets. Without explicit controls or ablations separating these factors, the difficulty estimates risk capturing surface distributions rather than multi-step reasoning load, undermining the core claim.

minor comments (1)

[Abstract] The abstract would be strengthened by including at least one key quantitative result (e.g., average improvement or correlation coefficient) to support the performance claims.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments. We address each major point below and outline the revisions we will make to strengthen the manuscript.

read point-by-point responses

Referee: [Abstract] Abstract: the central claims of 'consistent outperformance' and 'strong robustness' are asserted without any quantitative results, error bars, baseline implementation details, or statistical tests. This omission is load-bearing because it prevents verification of whether Q-DAPS actually improves on existing methods or merely reflects post-hoc choices.

Authors: We agree that the abstract would be strengthened by including key quantitative highlights. The full manuscript reports specific performance gains (e.g., entropy-based difficulty estimates outperforming readability, retrieval, and popularity baselines on TriviaQA, NQ, MuSiQue, and QASC), along with robustness checks across hyperparameters, question types, plausibility paradigms, and model sizes, plus human alignment results and statistical comparisons. We will revise the abstract to incorporate representative quantitative results, references to the experimental sections, and notes on the evaluation protocol. revision: yes
Referee: [Method] Method (entropy of plausibility scores): the assumption that this entropy isolates reasoning challenges posed to LLMs is not secured against confounds such as model calibration artifacts, training-data answer frequency, or lexical cues in the candidate sets. Without explicit controls or ablations separating these factors, the difficulty estimates risk capturing surface distributions rather than multi-step reasoning load, undermining the core claim.

Authors: We appreciate this concern about potential confounds. The manuscript already contains extensive ablations showing that Q-DAPS remains stable across multiple plausibility estimation paradigms, model sizes, and realistic settings, which partially addresses calibration and lexical issues by demonstrating consistency beyond single-model artifacts. Evaluations on four diverse datasets further reduce the risk of dataset-specific frequency biases. That said, we will add an explicit discussion subsection on these confounds, including any additional correlation analyses with answer frequency and lexical features, and note limitations where direct isolation is not feasible. revision: partial

Circularity Check

0 steps flagged

No circularity: Q-DAPS is a direct definition evaluated empirically

full rationale

The paper defines question difficulty directly as the entropy of plausibility scores over candidate answers and then reports empirical performance on four fixed QA datasets against baselines, plus robustness checks and human alignment. No step claims a derivation, prediction, or uniqueness result that reduces by construction to a fitted parameter, self-citation chain, or renamed input; the central quantity is introduced as a new signal rather than recovered from the evaluation data or prior author work. The derivation chain is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on abstract only: the central claim rests on the unstated assumption that plausibility scores from LLMs are a faithful proxy for reasoning difficulty and that entropy is the appropriate aggregation function. No explicit free parameters or invented entities are named in the abstract.

axioms (1)

domain assumption Plausibility scores computed by LLMs reflect the reasoning challenges of a question
Invoked implicitly when entropy of those scores is taken as the difficulty measure

pith-pipeline@v0.9.0 · 5499 in / 1026 out tokens · 54064 ms · 2026-05-13T05:24:02.177286+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
Q-DAPS estimates question difficulty by computing the entropy of plausibility scores over candidate answers... H(q) = −∑ DePls_norm_i × log2 DePls_norm_i ... Diff_q = H(q) / log2 N
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat recovery and J-cost orbit unclear
We hypothesize that the entropy of normalized plausibility scores provides an effective signal for estimating question difficulty.

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

[1]

InFindings of the Association for Computational Linguistics: ACL 2024, pages 14636–14650, Bangkok, Thailand

Measuring retrieval complexity in question answering systems. InFindings of the Association for Computational Linguistics: ACL 2024, pages 14636–14650, Bangkok, Thailand. Association for Computational Linguistics. Aaron Grattafiori, Abhimanyu Dubey, Abhinav Jauhri, Abhinav Pandey, Abhishek Kadian, Ahmad Al- Dahle, Aiesha Letman, Akhil Mathur, Alan Schelte...

work page 2024
[2]

How knowledge popularity influences and enhances llm knowledge boundary perception. OpenAI, Josh Achiam, Steven Adler, Sandhini Agarwal, Lama Ahmad, Ilge Akkaya, Florencia Leoni Ale- man, Diogo Almeida, Janko Altenschmidt, Sam Alt- man, Shyamal Anadkat, Red Avila, Igor Babuschkin, Suchir Balaji, Valerie Balcom, Paul Baltescu, Haim- ing Bao, and Others. 20...

work page 2024
[3]

Vatsal Raina and Mark Gales

Qwen2.5 technical report. Vatsal Raina and Mark Gales. 2024. Question difficulty ranking for multiple-choice reading comprehension. Vatsal Raina, Adian Liusie, and Mark Gales. 2023. As- sessing distractors in multiple-choice tests. InPro- ceedings of the 4th Workshop on Evaluation and Com- parison of NLP Systems, pages 12–22, Bali, Indone- sia. Associatio...

work page 2024
[4]

Applications of entropy in data analysis and machine learning: A review.Entropy, 26(12). C. E. Shannon. 1948. A mathematical theory of commu- nication.Bell System Technical Journal, 27(3):379– 423. C. Spearman. 1904. The proof and measurement of as- sociation between two things.The American Journal of Psychology, 15(1):72–101. Shuchang Tao, Liuyi Yao, Han...

work page 1948
[5]

Gemma Team, Morgane Riviere, Shreya Pathak, Pier Giuseppe Sessa, Cassidy Hardin, Surya Bhu- patiraju, Léonard Hussenot, Thomas Mesnard, and Others

Gemma 3 technical report. Gemma Team, Morgane Riviere, Shreya Pathak, Pier Giuseppe Sessa, Cassidy Hardin, Surya Bhu- patiraju, Léonard Hussenot, Thomas Mesnard, and Others. 2024. Gemma 2: Improving open language models at a practical size. Katherine Tian, Eric Mitchell, Allan Zhou, Archit Sharma, Rafael Rafailov, Huaxiu Yao, Chelsea Finn, and Christopher...

work page 2024
[6]

Finally,Difficultymeasures how challenging a question is to answer, ranging from straightforward fact-based queries (Padó, 2017) to complex reason- ing problems (Xu et al., 2024)

or to request clarification (Chi et al., 2024). Finally,Difficultymeasures how challenging a question is to answer, ranging from straightforward fact-based queries (Padó, 2017) to complex reason- ing problems (Xu et al., 2024). A.1 Difficulty Question difficulty can be assessed from differ- ent perspectives. One common perspective fo- cuses onreadability....

work page 2024
[7]

As shown in Table 8, there are no duplicate answers

No Duplicates:We compare the generated candidate answers using a semantic equality function. As shown in Table 8, there are no duplicate answers

work page
[8]

Table 8 confirms this con- dition is satisfied

Range of Plausibility Scores:We verify that all plausibility scores fall within the valid range of 0 to 100. Table 8 confirms this con- dition is satisfied

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[9]

Table 8 confirms this requirement is met

Number of Candidate Answers:We check that the LLM produced exactly 20 candidate answers as requested. Table 8 confirms this requirement is met. B.2 Popularity Debiasing In this stage, we employ the HintEval toolkit (Mozafari et al., 2025c) to compute the popularity 8 of each candidate answer. The Popularitycomponent retrieves the popularity scores from Wi...

work page 1948
[10]

is a compact yet capable model optimized for fast inference and moderate computational ef- ficiency.Gemma 3 4B(Team et al., 2025) is a lightweight transformer model designed for effi- cient language processing while maintaining com- petitive performance on standard NLP benchmarks. Medium LMsMistral 7B(Jiang et al., 2023) is a high-performance, decoder-onl...

work page 2025
[11]

Yes" or

is an improved version of LLaMA designed for enhanced reasoning and generalization in lan- guage tasks. Large LMsMistral 24B(Jiang et al., 2023) is a larger variant designed for high-capacity reason- ing and advanced generative performance, with a strong ability to follow complex instructions. Gemma 2 27B(Team et al., 2024) is a mid-scale model offering a...

work page 2023
[13]

Candidate Answer

A detailed explanation of the reasoning behind the plausibility score. Format your response as a JSON object, where the candidate is represented as: [ { "Candidate Answer": "<candidate_answer>", "PlausibilityScore": <plausibility_score>, "Justification": "<justification>" } ] The output must be a valid JSON object only. Figure 10: The placeholder <questio...

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[14]

A non-zero plausibility score as a number between 0 and 100

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[15]

Candidate Answer

A detailed explanation of the reasoning behind the plausibility score. Format your response as a JSON list, where each candidate is represented as: [ { "Candidate Answer": "<candidate_answer>", "PlausibilityScore": <plausibility_score>, "Justification": "<justification>" } ] The output must be a valid JSON list only. Figure 16: The placeholder <question> ...

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[16]

shown in Figure 3

We use our standard prompt for generating candidate answers and plausibility scores, as 10We use 10 candidates for clearer visualization and analy- sis compared to all 20. shown in Figure 3. This prompt explicitly instructs the LLM not to include the correct answer as a candidate, while still providing it in the prompt to guide plausibility estimates

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[17]

The modified prompt used for this setting is shown in Figure 16

We modify the prompt to remove the instruc- tion excluding the correct answer, allowing the LLM to rely on its knowledge to iden- tify and potentially include the correct answer among the candidates. The modified prompt used for this setting is shown in Figure 16. To ensure consistency and reproducibility, we set the temperature to zero in both scenarios,...

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