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arxiv: 2605.28264 · v1 · pith:LETX6BEXnew · submitted 2026-05-27 · 💻 cs.AI

Entropy Distribution as a Fingerprint for Hallucinations in Generative Models

Mattia J. Villani , Pranav Deshpande , Akshay Seshadri , Romina Yalovetzky , Niraj Kumar This is my paper

Pith reviewed 2026-06-29 12:08 UTC · model grok-4.3

classification 💻 cs.AI
keywords hallucination detectiontoken entropy distributioncalibrated entropy scoresingle forward passblack-box accesslarge language modelsstatistical hypothesis test
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The pith

The distribution of token-level entropies serves as a fingerprint for hallucinations, enabling single-pass detection via a calibrated score that combines mean and maximum signals.

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

The paper argues that hallucinations leave a signature in the full spread of token entropies, not merely their average as captured by perplexity. Shape and tail behavior of this distribution carry independent information that allows formal detection as a hypothesis test. The authors introduce the Calibrated Entropy Score, which normalizes mean and peak entropy against a reference cumulative distribution to produce comparable outputs across models. The method needs only one forward pass and black-box logit access, yet supplies finite-sample calibration bounds and exponential convergence of detection probability with length. Empirical results across eight benchmarks and ten models show it matches the accuracy of far more expensive multi-sample approaches.

Core claim

We provide theoretical background and empirical evidence that the distribution of token-level entropies, beyond the mean captured by perplexity or length-normalised entropy, serves as a fingerprint of hallucination, with distributional shape and tail behaviour carrying independent signal. We formalize hallucination detection as a statistical hypothesis test and propose the Calibrated Entropy Score (CES), a lightweight algorithm requiring only a single forward pass and black-box access to token logits. CES combines the mean signal with the maximum signal of the generated entropy through a calibrated reference CDF, producing scores that are directly comparable across models and tasks. We estab

What carries the argument

The Calibrated Entropy Score (CES), formed by mapping the mean and maximum of the per-token entropy sequence onto a reference cumulative distribution function to yield a comparable hallucination score.

If this is right

  • CES achieves the highest detection performance among single-pass black-box methods on eight QA benchmarks and ten generator models.
  • Finite-sample calibration guarantees are provided by a random-length Dvoretzky--Kiefer--Wolfowitz inequality.
  • Detection probability converges exponentially to one as generation length grows.
  • Produced scores are directly comparable across open-source and API-access models without retraining.

Where Pith is reading between the lines

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

  • The approach could be inserted into decoding loops to steer away from high-entropy token sequences in real time.
  • A broader reference CDF built from mixed domains might remove the need for task-specific recalibration.
  • If tail behavior dominates, then monitoring only the highest-entropy tokens could yield cheaper approximations.

Load-bearing premise

A single reference CDF can calibrate the mean and maximum entropy signals so that the resulting scores stay comparable and the detection guarantees remain valid across different models, tasks, and generation lengths.

What would settle it

A controlled test on a held-out model and task showing that CES scores lose calibration or fail to separate hallucinated from correct outputs at rates predicted by the exponential convergence bound.

Figures

Figures reproduced from arXiv: 2605.28264 by Akshay Seshadri, Mattia J. Villani, Niraj Kumar, Pranav Deshpande, Romina Yalovetzky.

Figure 1
Figure 1. Figure 1: Calibrated Entropy Score (CES) hallucination detection. A single forward pass produces token distributions p (t) , from which entropies h (t) are computed. Two summary statistics (the mean entropy h¯ and the maximum entropy hmax) are mapped through the calibrated CDF Fb0 and combined via a geometric mean. The reference CDF, Fb0, is estimated offline from an oracle-labeled calibration set (dashed path; Algo… view at source ↗
Figure 2
Figure 2. Figure 2: illustrates the separation. Panel (a) shows representative ECDFs for a median-effect-size experiment, where faithful and hallucinated entropy distributions are visibly offset. Panel (b) displays the distribution of dKS distances across all 80 experiments, with the rejection threshold annotated. Panel (c) shows that the mean-centred shape signal is universally significant, confirming distributional shape di… view at source ↗
Figure 3
Figure 3. Figure 3: Benchmark comparison of confabulation detection methods across all model–dataset experiments. (a) Average rank (lower is better) for each method, aggregated over all 80 open-weight and API model experiments. Bar annotations indicate the average rank and number of first-place finishes. (b) Nemenyi Critical Difference (CD) diagram following [7]. CES (unsupervised) ranks among the top clique without requiring… view at source ↗
Figure 4
Figure 4. Figure 4: Per-experiment AUROC for all 17 methods across 10 models × 8 datasets. Each bar represents one method’s AUROC; models are grouped by row and datasets by column. CES (unsupervised, starred) achieves a median AUROC of 0.653, comparable to Embedding Regression (0.665) and KLE variants (0.647–0.651). Setup. We evaluate 7 detection methods across our full 10 × 8 experimental grid: CES (supervised and unsupervis… view at source ↗
Figure 5
Figure 5. Figure 5: Token entropy autocorrelation analysis across 18,705 sequences. (a) Distribution of lag-1 ACF; median ρ1 = 0.061, well below the threshold for concern. (b) Lag-1 vs. lag-2 autocorrelation scatter; second-order dependence is weaker (median ρ2 ≈ 0.03). (c) Effective sample size ratio neff/n; most sequences retain >80% effective samples. Only 15.8% of sequences have |ρ1| > 0.3, yielding substantially reduced … view at source ↗
Figure 6
Figure 6. Figure 6: KS test power vs. sample size across all 80 experiments (10 models × 8 datasets). Each cell shows the log10 p-value as a function of subsample size for one model–dataset pair. Significance threshold (α=0.05) is shown as a horizontal line. Power increases monotonically with sample size; 65% of all 3,200 resamples achieve significance. E.5 Per-Dataset ECDF Gallery [PITH_FULL_IMAGE:figures/full_fig_p030_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Per-dataset ECDF gallery. Faithful (blue) vs. hallucinated (red) entropy distributions for each of 8 datasets, pooled across all 10 models. dKS and p-values are annotated per panel. Visual separation directly corresponds to detection performance: datasets with clearly offset curves (SVAMP, TriviaQA) yield the highest AUROC. E.6 Generation Length and Test Power Setup. The KS test’s power depends on sample s… view at source ↗
Figure 8
Figure 8. Figure 8 [PITH_FULL_IMAGE:figures/full_fig_p032_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Combinatorial statistic ranking. (top 10 of 44 variants). Ranked by average rank across 80 experiments (lower = better). The CES formula geom(mean, max) achieves the best average rank (10.3), confirming that combining location (mean) and tail (max) entropy statistics via geometric aggregation optimally captures both distributional features. E.8 Calibration Contamination Setup. The supervised CES variant us… view at source ↗
Figure 10
Figure 10. Figure 10 [PITH_FULL_IMAGE:figures/full_fig_p033_10.png] view at source ↗
Figure 11
Figure 11. Figure 11 [PITH_FULL_IMAGE:figures/full_fig_p034_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Nemenyi Critical Difference diagram for 17 methods on 80 experiments. Methods connected by a thick bar are not significantly different (α = 0.05, CD = 2.78). CES (unsupervised) at rank 6.29 is within the top clique and statistically indistinguishable from the best-ranked methods (KLE (heat) at 6.16). 35 [PITH_FULL_IMAGE:figures/full_fig_p035_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: CES pairwise win rates against each of 16 benchmark methods. CES wins 854/1279 pairwise comparisons (66.8%) overall and beats 12/16 methods at >50% win rate. Median AUROC advantage (∆) annotated for each opponent. E.11 API vs. Open-Weight Model Comparison Setup. A key practical question is whether entropy-based detection generalises to API-only models where only token-level log-probabilities (not full log… view at source ↗
Figure 14
Figure 14. Figure 14: Per-dataset CES AUROC comparison: API (red) vs. open-weight (blue) models. Significant differences (p < 0.05) are marked with asterisks. 4/8 datasets show statistically significant differences, with the direction varying by task type. 0.4 0.5 0.6 0.7 0.8 0.9 CES (supervised) score 0 1 2 3 4 5 6 7 Density (a) CES Score Distributions Open-Weight (n=21559) API (n=14924) Open-Weight API 0.0 0.2 0.4 0.6 0.8 1.… view at source ↗
Figure 15
Figure 15. Figure 15: Distributional properties: API vs. open-weight models. (a) AUROC distributions overlap substantially. (b) Cohen’s d distributions are comparable (API median 0.494, open 0.515). (c) Dataset difficulty correlation between model types (Pearson r = −0.49), suggesting that difficult datasets for API models are not necessarily difficult for open-weight models and vice versa. 37 [PITH_FULL_IMAGE:figures/full_fi… view at source ↗
Figure 16
Figure 16. Figure 16: Method ranking comparison: internal methods ranked by median AUROC separately for API and open-weight models. Spearman rank correlation ρ = 0.673 (p = 0.098) indicates moderate agreement in method effectiveness across model types. E.12 Empirical Verification of CES Error Bounds Setup. Theorem 5 establishes that the Type I and Type II errors of the CES test decay exponentially with generation length m. Spe… view at source ↗
Figure 17
Figure 17. Figure 17: Empirical verification of CES error bounds (Theorem 5). Type I error (false positive rate) and Type II error (false negative rate) as a function of generation length m, measured on synthetic i.i.d. generations drawn from pooled faithful and hallucinated token entropy distributions. Dashed lines show the theoretical exponential upper bounds. Both error rates decay exponentially with m, and the empirical cu… view at source ↗
read the original abstract

Large Language Models (LLMs) often generate factually incorrect outputs, commonly termed hallucinations, that undermine trust and limit deployment in high-stakes settings. Existing hallucination detection methods typically require multiple forward passes, or access to model internals. In this work, we provide theoretical background and empirical evidence that the distribution of token-level entropies, beyond the mean captured by perplexity or length-normalised entropy, serves as a fingerprint of hallucination, with distributional shape and tail behaviour carrying independent signal. We formalize hallucination detection as a statistical hypothesis test and propose the Calibrated Entropy Score (CES), a lightweight algorithm requiring only a single forward pass and black-box access to token logits. CES combines the mean signal with the maximum signal of the generated entropy through a calibrated reference CDF, producing scores that are directly comparable across models and tasks. We establish finite-sample calibration guarantees via a novel random-length Dvoretzky--Kiefer--Wolfowitz inequality, and also prove that CES detects hallucinations with probability converging to one exponentially fast in the generation length. Across eight QA benchmarks and ten generator models spanning open-source and API access models, CES achieves the highest detection performance among all single-pass black-box methods while providing formal error guarantees that existing heuristics lack. Remarkably, CES is statistically indistinguishable from multi-sample methods that require far greater computational cost, closing the gap between lightweight and expensive detection and making it suitable for real-time, large-scale deployment.

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 claims that the distribution of token-level entropies (shape and tails beyond the mean) serves as a fingerprint for hallucinations in LLMs. It formalizes detection as a hypothesis test and introduces the Calibrated Entropy Score (CES), a single-pass black-box method that combines mean and max entropy signals via a calibrated reference CDF. Theoretical contributions include finite-sample guarantees from a novel random-length Dvoretzky–Kiefer–Wolfowitz inequality and exponential convergence of detection probability with generation length. Empirically, CES outperforms other single-pass methods on eight QA benchmarks across ten models (open-source and API) and is statistically indistinguishable from multi-sample baselines.

Significance. If the single-reference-CDF calibration transfers without model-specific bias, the result would be significant: it supplies the first lightweight detector with explicit finite-sample and exponential-rate guarantees, closing the performance gap to expensive multi-sample methods while remaining deployable at scale.

major comments (2)
  1. [Abstract and §3] Abstract and §3 (CES definition): the central cross-model claim requires that one fixed reference CDF calibrates both mean and max entropy signals so that CES remains comparable and the DKW-based finite-sample guarantees hold for all ten models. Token-level entropy distributions are shaped by architecture, scale, tokenizer, and training objective; if these induce different families, the quantile mapping introduces model-specific bias that invalidates both the hypothesis-test interpretation and the claim of matching multi-sample methods without per-model recalibration.
  2. [Theoretical development] Theoretical development (novel random-length DKW inequality): the inequality assumes the underlying distribution is fixed, yet the cross-model empirical claims implicitly treat the reference CDF as universal. No evidence is provided that the inequality’s assumptions survive the shift from reference-construction models to the ten evaluation models.
minor comments (2)
  1. Clarify whether the reference CDF was constructed from a held-out set of models/tasks independent of the ten evaluated generators.
  2. Add explicit statistical tests (e.g., paired significance) for the claim that CES is indistinguishable from multi-sample methods.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive comments on the cross-model calibration of CES and the scope of the theoretical guarantees. We address each major comment below.

read point-by-point responses

  1. Referee: [Abstract and §3] Abstract and §3 (CES definition): the central cross-model claim requires that one fixed reference CDF calibrates both mean and max entropy signals so that CES remains comparable and the DKW-based finite-sample guarantees hold for all ten models. Token-level entropy distributions are shaped by architecture, scale, tokenizer, and training objective; if these induce different families, the quantile mapping introduces model-specific bias that invalidates both the hypothesis-test interpretation and the claim of matching multi-sample methods without per-model recalibration.

    Authors: Empirical results across ten diverse models (open- and closed-source) and eight benchmarks show that a single reference CDF yields comparable CES values and performance statistically indistinguishable from multi-sample baselines without per-model recalibration. This supports the practical validity of the hypothesis-test framing. We will revise §3 to detail the reference CDF construction from a multi-model calibration set and add a brief discussion of observed robustness to architectural differences. revision: partial

  2. Referee: [Theoretical development] Theoretical development (novel random-length DKW inequality): the inequality assumes the underlying distribution is fixed, yet the cross-model empirical claims implicitly treat the reference CDF as universal. No evidence is provided that the inequality’s assumptions survive the shift from reference-construction models to the ten evaluation models.

    Authors: The random-length DKW inequality is applied to the fixed reference CDF once constructed, supplying finite-sample guarantees for quantile estimation on subsequent generations from any model. While a formal transfer theorem under arbitrary model shift is not derived, the evaluation includes models held out from reference construction, and the observed performance provides supporting empirical evidence. We will add a clarifying paragraph in the theoretical section on the conditional nature of the guarantees and the role of the empirical validation. revision: partial

Circularity Check

0 steps flagged

No circularity; derivation relies on novel inequality and independent statistical formalization

full rationale

The paper formalizes detection as a hypothesis test, defines CES via mean/max entropy signals combined through a reference CDF for cross-model comparability, and derives finite-sample guarantees plus exponential convergence from a novel random-length DKW inequality. No quoted step reduces a claimed prediction or result to a fitted input by construction, nor does any load-bearing premise collapse to self-citation or ansatz smuggling. The reference CDF construction is not shown to be data-dependent in a way that forces the detection claims, and the theoretical results are presented as independent of the empirical benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the premise that hallucination status systematically alters the entropy distribution in a calibratable way; the reference CDF is the main unstated element whose construction is not detailed in the abstract.

free parameters (1)
  • reference CDF construction parameters
    The reference cumulative distribution function used for calibration is necessarily chosen or fitted from some data or model outputs to enable cross-model comparability.
axioms (1)
  • domain assumption Hallucinated generations exhibit distinguishable entropy distribution shape and tail behaviour compared to factual generations in a manner consistent enough for a single reference CDF to calibrate across models and tasks.
    This is the load-bearing premise that allows the fingerprint and the CES guarantees to hold.

pith-pipeline@v0.9.1-grok · 5809 in / 1401 out tokens · 54215 ms · 2026-06-29T12:08:26.155191+00:00 · methodology

discussion (0)

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Reference graph

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