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REVIEW 4 major objections 5 minor 1 cited by

Hallucination Detection: A Probabilistic Framework Using Embeddings Distance Analysis

T0 review · 4 major / 5 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read The paper claims that hallucinated and genuine LLM responses are structurally distinct in embedding space, and that a probabilistic distance-based classifier can flag hallucinations at up to 66% accuracy without external knowledge.

desk verdict Mechanically careful but confounded: the central structural-difference claim reduces to 'Llama2 and Llama3 differ in embedding distances,' so the paper needs a same-model control before it can be taken seriously. read the letter →

arxiv 2502.08663 v1 pith:H5MZHWZ6 submitted 2025-02-10 cs.CL cs.AIcs.CY

Emanuele Ricco , Lorenzo Cima , Roberto Di Pietro This is my paper · ORCID
classification cs.CLcs.AIcs.CY MSC 68T5062H3062G07
keywords LLMhallucinationembeddingdistanceMinkowskidetectionkerneldensityestimationBERTembeddingsstructuraldifference
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

This paper claims that hallucinated and genuine LLM responses are structurally distinct in embedding space, and that the distinction can power a detector with no external knowledge or fact-checking. The authors generate 64 questions about events between September 2022 and September 2023, collect multiple answers from Llama2 (training cutoff September 2022) and Llama3 (cutoff December 2023), and compare the distributions of pairwise Minkowski distances among the BERT embeddings of each response's extracted keywords. They report a statistically significant difference between hallucinated and non-hallucinated distance distributions (Wilcoxon test, $p < 0.01$) that persists across every tested norm ($p \in \{0.5, 1, 2\}$), keyword count ($n \in \{1, \ldots, 10\}$), and response count ($r \in \{4, \ldots, 16\}$). A probabilistic classifier built on these distributions reaches 66% accuracy in its best configuration. If the structural claim holds, hallucination detection becomes a geometry problem rather than a fact-checking problem.

What carries the argument

The load-bearing object is the Minkowski distance between BERT embeddings of response keywords. Each response is reduced by KeyBERT to its $n$ most important keywords ($n = 1, \ldots, 10$), those keywords are embedded into 768-dimensional BERT vectors, and all pairwise distances within each class are computed under three Minkowski norms ($p \in \{0.5, 1, 2\}$). Those pairwise distances become the unit of analysis: KL divergence and median difference quantify the gap between hallucinated and non-hallucinated distance distributions, the Wilcoxon test establishes that the gap is statistically significant, and Gaussian kernel density estimation converts the training distributions into likelihood models. At test time, each new response's distances to all training responses of each class are scored under the two KDE log-likelihoods, and the class with the higher summed log-likelihood wins. The same machinery yields both the claimed scale-free structural result and the detector.

What would settle it

Run the identical distance analysis within a single model on verifiable correct and incorrect answers from the same generator; if the two distance distributions do not separate under the same Wilcoxon comparison, the reported structural difference is a property of the two generators, not of hallucination itself.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that hallucinated content carries a measurable structural signature: the pairwise embedding distances of hallucinated responses and those of correct responses come from different distributions, and the difference is large enough to be statistically detectable and stable across parameters. The claim is grounded in an artificial dataset in which Llama2 necessarily hallucinates answers about a year (September 2022 to September 2023) beyond its training cutoff, while Llama3, whose training covers that year, produces correct answers. Distances are computed between BERT embeddings of the $n$ most important keywords extracted from each response, using Minkowski norms $p = 0.5$, $p = 1$, and $p = 2$. The authors report that the distributional gap, measured by KL divergence and median difference, widens as the number of keywords and responses grows, remains significant under a Wilcoxon test for every configuration, and is scale-free in the sense that the qualitative separation does not depend on the norm, the keyword count, or the number of responses. The paper presents this as the first demonstration that hallucination has a geometric structure in embedding space.

Load-bearing premise

The labels that split responses into hallucinated and genuine come entirely from which model generated them: Llama2 is assumed to hallucinate every answer about the post-cutoff year, and Llama3 is assumed to answer every such question correctly.

Editorial extensions

If this is right

  • A hallucination detector can operate on response geometry alone, needing no external knowledge base, retriever, or fact-checker.
  • Because the separation is claimed for every tested norm, keyword count, and response count, a deployment can trade cost for accuracy without expecting the signal to disappear.
  • Fractional distances ($p = 0.5$) amplify the distribution gap and produced the best accuracy (0.66 at $r=8$, $n=1$), pointing to high-dimensional distance behavior as a tuning lever.
  • The accuracy comparison against published detectors places the geometry-only approach above token-probability and entropy baselines but below HaloScope on the reported TruthfulQA numbers.
  • The cutoff-gap data-generation recipe, asking about the interval covered by one model's training but not another's, scales to any pair of models and can produce labeled hallucination data without manual annotation.

Reading between the lines

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

  • If the structural separation is genuine and not just a between-model artifact, it suggests hallucinated text is produced by a different statistical regime within a model, not merely text that happens to be false; that would make embedding geometry a useful probe for interpretability work.
  • A decisive extension is to repeat the exact distance protocol within a single model, comparing factually correct answers with factually wrong answers from the same generator; this would separate the hallucination signal from model style.
  • The 66% ceiling with a simple KDE rule suggests headroom: combining the distance score with semantic entropy or self-evaluation could yield a stronger joint detector than any single signal.
  • Testing the same claims with other embedding models (not only BERT) and other model families would show whether the scale-free structural difference transfers or is tied to these specific encoders.
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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

4 major / 5 minor

Summary. The paper proposes a probabilistic framework for detecting LLM hallucinations by analyzing Minkowski distances between BERT embeddings of keywords extracted from model responses. The authors construct a synthetic dataset using Llama2 and Llama3, labeling Llama2 responses to questions about the September 2022–September 2023 period as hallucinated and Llama3 responses as non-hallucinated, purely based on training cutoff dates. They then compute intra-class distance distributions, report statistically significant differences via Wilcoxon tests and KL divergence, claim these differences are scale-free across distance norms, keyword counts, and response counts, and build a KDE-based classifier that achieves 66% accuracy for one configuration. The paper's central claim is that this work is the first to show structural differences between hallucinated and correct LLM responses.

Significance. If the central claim were valid, a purely geometric, external-knowledge-free signal for hallucination would be practically valuable and could support a simple and cheap detection method. The paper also provides a concrete, reproducible pipeline (questions, models, embeddings, and statistical tests) that could serve as a baseline for embedding-distance-based hallucination detection. However, the significance of the claimed results is entirely undermined by the experimental design: the ground-truth labels are perfectly confounded with model identity, so every reported distributional difference and the classifier's accuracy may reflect model-specific style, vocabulary, or decoding behavior rather than hallucination itself. Because the appendix examples further contradict the label definitions, the paper does not establish the existence of a structural signal specific to hallucinated content.

major comments (4)
  1. [§3.2, Dataset] The dataset labels are assigned solely by generator identity. The authors write, "This choice ensures that the answers generated by Llama2 hallucinated, while those generated by Llama3 did not," but this equates model identity with hallucination status: Llama2 responses are the 'hallucinated' class and Llama3 responses are the 'non-hallucinated' class for every question. Consequently, every distance-distribution difference reported in §4.1 and used in §3.4 is confounded with model-specific properties (e.g., token distributions, response length, decoding style) after BERT embedding. The caveat in §5.2 that "a little subset of Llama3 responses might be hallucinated as well" weakens the labels further but does nothing to remove the perfect correlation between class and model; no same-model comparison is offered.
  2. [§3.4 and §4.2, Test] The detection experiment shares both the same 64 questions and the same two generating models between training and test. The KDE likelihood scoring therefore has access to a training set in which the two classes are exactly the two models, so a classifier can separate the test responses by reproducing the Llama2/Llama3 embedding geometry rather than by detecting hallucination. The 66% accuracy in Table 3 is thus not evidence for a structural signal specific to hallucinated content. A control experiment using a single model with verified factual errors versus correct responses from that same model would be required to support the paper's central claim.
  3. [Appendix A.2, Question vs. Responses] The qualitative examples contradict the label definitions. NH-1 asserts, among other things, that COVID-19 caused economic decline and that a phone call between President Trump and Chinese Vice Premier Liu He eased tariffs, claims that are false for the September 2022–September 2023 period; NH-2 is a nonsensical enumeration of countries. Conversely, H-1 is an unrelated assignment-style prompt and H-2 is a collage of motivational quotations, neither of which is a fluent, factually wrong response to the question about the global economy. These examples show that the automatic labels do not track the intended construct of hallucination versus correct content.
  4. [§4.2, Table 4, Comparison with state of the art] The baseline accuracies are copied from Du et al. (2024), where they were reported on TruthfulQA, while the proposed method is evaluated on the authors' synthetic Llama2/Llama3 dataset. The two evaluation settings differ in topic distribution, label source, and response length, so the table does not support the statements that the method is "second" or "comparable with the best results in the field." The baselines need to be run on the same evaluation protocol and the same test split before any comparison is meaningful.
minor comments (5)
  1. [§4.1] The Wilcoxon tests are performed on hundreds of thousands of paired distances per configuration, so p<0.01 carries little information; report an effect size (e.g., rank-biserial correlation) and a confidence interval.
  2. [§4.1 and Appendix A.4] The "scale-free" property is inferred from visual similarity across boxplots; state a quantitative invariance criterion (e.g., consistent sign and approximate proportionality of median differences across conditions) and test it explicitly.
  3. [§3.2] Clarify whether the 4-bit quantization applies to model weights or to the generation process, and report the generation temperature, top-p, top-k, and maximum token settings; reproducibility requires these details.
  4. [§3.4] The score comparison "Shall > Snohall" is equivalent to comparing average log-likelihoods only because the two sums have equal length qr; state this equivalence explicitly to avoid confusion.
  5. [Typos and wording] Typos and wording issues include: Table 1 caption "Minkoswki" should be "Minkowski"; §6 uses "hallucinationed" twice; reference Skala (2013) title contains "Oexpected"; and §4.1's sentence about the 279% KL increase lacks the baseline it increases with respect to.

Circularity Check

2 steps flagged · score 6.0 of 10

Hallucination labels are defined by model identity, so the 'structural difference' and the 66% detector reduce to comparing Llama2 versus Llama3 instead of hallucination per se.

  1. self definitional [Section 3.2, Dataset]
    "This choice ensures that the answers generated by Llama2 hallucinated, while those generated by Llama3 did not."

    The central claim is that hallucinated content is structurally different from correct content. However, hallucination status is not independently established; it is stipulated to mean 'generated by Llama2' while non-hallucinated means 'generated by Llama3'. Every distance distribution compared in Section 4.1 is therefore a distribution of Llama2 versus Llama3 response embeddings. The claimed structural difference between hallucinated and genuine responses reduces, by the paper's own construction, to the structural difference between two different models. Stylistic, lexical, or decoding differences between Llama2 and Llama3 would produce the same result, so the conclusion about hallucination is indistinguishable from the input model identities.

  2. fitted input called prediction [Section 3.4, Test; Section 4.2, Test results]
    "we referred to the (qr 2) distances within each class fitting probability densities using Gaussian Kernel Density Estimation (KDE), separately for hallucinated and non-hallucinated data"

    The detector's class-conditional densities are fitted to the same two models and the same 64 questions that generate the test responses, and the labels are the same model-identity labels fixed in Section 3.2. Thus the hallucinated density is effectively a Llama2 density and the non-hallucinated density is a Llama3 density. The decision rule Class = 1 if Shall > Snohall is therefore a rule for choosing which model produced a test embedding. The reported 66% accuracy is an estimate of how well embedding distances separate Llama2 from Llama3, relabelled as hallucination detection.

full rationale

The paper's internal statistics are coherent: it builds KDE models, computes distances, runs Wilcoxon tests, and reports held-out accuracy. The circularity lies one level above the statistics. The class labels are not grounded in factuality checks; they are assigned by generator identity in Section 3.2. Consequently, the first contribution ('hallucinated content has structural differences') is an empirical statement about Llama2 versus Llama3 embedding-distance distributions, and the detector's accuracy measures separability of the two model families on the same 64 questions. This is a self-definitional reduction of the target construct, not a mere confound, because the paper explicitly equates hallucination with Llama2 output and correctness with Llama3 output. There is no load-bearing self-citation chain or imported uniqueness theorem; the issue is the operational definition of hallucination. The appendix labels further weaken the construct: the 'non-hallucinated' response NH-1 contains dubious assertions, while 'hallucinated' responses include an unrelated assignment prompt and a quote collage, which are not typical factually wrong answers. The limitations section acknowledges that some Llama3 responses might be hallucinated, but a partially noisy label is not sufficient to break the model-identity equivalence. For these reasons, the central claim and the detection result reduce by construction to model identification, warranting a partial circularity score of 6.

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

The paper introduces no new theoretical entities. The core assumptions are all domain assumptions about the validity of the model-identity-based labels and about the representativeness of the synthetic dataset. The main free parameter is the post hoc selection of the best configuration, which inflates the reported accuracy.

free parameters (3)
  • Best configuration selection (r=8, n=1, p=0.5) = r=8, n=1, p=0.5
    The reported 66% accuracy is the maximum across 7 r values, 10 n values, and 3 p values (210 configurations), chosen post hoc on the test set without multiple-testing correction.
  • KDE bandwidth = Not specified
    The Gaussian KDE requires a bandwidth parameter, but the paper does not state how it was selected; the resulting densities and classifications depend on it.
  • Number of questions q=64 = 64
    The dataset size is arbitrary and the 64 GPT-4 generated questions are not validated as a representative sample of hallucination triggers.
assumptions (4)
  • domain assumption Llama2 responses to post-cutoff questions are hallucinations and Llama3 responses are correct
    This is the sole source of ground-truth labels; it conflates model identity with hallucination status and is stated in Section 3.2.
  • domain assumption The distance distributions from these two models are representative of hallucinated vs. correct content in general
    The scale-free claim in Section 4.1 generalizes from this single synthetic setup without external validation.
  • domain assumption BERT embeddings and KeyBERT keywords capture the semantic content relevant to hallucination
    Invoked in Section 3.2 without validation against other embedding methods or keyword extraction techniques.
  • domain assumption Minkowski distances between keyword embeddings within a class follow a smooth density estimable by Gaussian KDE
    Used in Section 3.4; no justification is given for this density model or for the choice of Gaussian kernel.

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Cite this review

Pith. "Pith review of Hallucination Detection: A Probabilistic Framework Using Embeddings Distance Analysis." pith.science (2026). https://pith.science/paper/H5MZHWZ6

@misc{pith2026250208663,
  author       = {Pith},
  title        = {Pith review of: Hallucination Detection: A Probabilistic Framework Using Embeddings Distance Analysis},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/H5MZHWZ6}},
  note         = {Machine review of arXiv:2502.08663}
}
read the original abstract

Hallucinations are one of the major issues affecting LLMs, hindering their wide adoption in production systems. While current research solutions for detecting hallucinations are mainly based on heuristics, in this paper we introduce a mathematically sound methodology to reason about hallucination, and leverage it to build a tool to detect hallucinations. To the best of our knowledge, we are the first to show that hallucinated content has structural differences with respect to correct content. To prove this result, we resort to the Minkowski distances in the embedding space. Our findings demonstrate statistically significant differences in the embedding distance distributions, that are also scale free -- they qualitatively hold regardless of the distance norm used and the number of keywords, questions, or responses. We leverage these structural differences to develop a tool to detect hallucinated responses, achieving an accuracy of 66\% for a specific configuration of system parameters -- comparable with the best results in the field. In conclusion, the suggested methodology is promising and novel, possibly paving the way for further research in the domain, also along the directions highlighted in our future work.

Figures

Figures reproduced from arXiv: 2502.08663 by the authors.

Figure 2
Figure 2. Overview of the methodology implemented to extract information from the training responses. First, we created training keyword datasets, unique for each combination of responses (r) and keywords (n). So for each r value, we generated 10 distinct datasets, each con￾taining qr responses from Llama2 and Llama3 models. We then calculated the Minkowski distances reported in [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Overview of the methodology implemented to classify test responses. followed the same steps as training. Initially, we processed test responses to extract keywords, and then we created test keyword datasets for each combination of r and n, generat￾ing embeddings for both hallucinated and non-hallucinated responses. Once we obtained the embeddings, we computed distances between each test embedding and all training em… view at source ↗
Figure 4
Figure 4. Training distributions of Minkowski distances with r = 16. Statistical significance: ***: p < 0.01, **: p < 0.05, *: p < 0.1. for further, future research. 4.2. Test For the test phase, we evaluated two key metrics across all combinations for both r and n parameters fixing p = 2.0, as shown in [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figures from the paper (10 more)
Figure 5
Figure 5. Figure 5: KL Divergence and median Difference vs n at different r for p = 2.0 using training data. itives and true negatives), while F1 score for hallucinated responses only considers how well the model identifies hal￾lucinations. General accuracy, in particular, is better with …
Figure 6
Figure 6. Figure 6: Model accuracy and hallucination F1 score vs r at different n for p = 2.0 a sentence is genuine or hallucinated. We evaluated this methodology for many different parameter configurations, considering both the general accuracy of the model and a specific F1 score for ha…
Figure 7
Figure 7. Figure 7: Training distributions of Minkowski distances with r = 4. n = 1 n = 2 n = 3 n = 4 n = 5 n = 6 n = 7 n = 8 n = 9 n = 10 p = 0.5 0.092 0.008 0.073 0.069 0.093 0.140 0.166 0.175 0.247 0.216 p = 1.0 0.094 0.008 0.078 0.071 0.104 0.143 0.165 0.172 0.254 0.212 p = 2.0 0.110 …
Figure 8
Figure 8. Figure 8: Training distributions of Minkowski distances with r = 6. n = 1 n = 2 n = 3 n = 4 n = 5 n = 6 n = 7 n = 8 n = 9 n = 10 p = 0.5 0.055 0.015 0.066 0.071 0.075 0.109 0.096 0.076 0.097 0.095 p = 1.0 0.060 0.017 0.065 0.071 0.078 0.105 0.096 0.081 0.096 0.097 p = 2.0 0.075 …
Figure 9
Figure 9. Figure 9: Training distributions of Minkowski distances with r = 8. n = 1 n = 2 n = 3 n = 4 n = 5 n = 6 n = 7 n = 8 n = 9 n = 10 p = 0.5 0.082 0.035 0.099 0.056 0.056 0.085 0.078 0.061 0.082 0.095 p = 1.0 0.092 0.036 0.101 0.058 0.057 0.088 0.079 0.060 0.081 0.100 p = 2.0 0.121 …
Figure 10
Figure 10. Figure 10: Training distributions of Minkowski distances with r = 10. n = 1 n = 2 n = 3 n = 4 n = 5 n = 6 n = 7 n = 8 n = 9 n = 10 p = 0.5 0.073 0.024 0.080 0.070 0.067 0.116 0.101 0.092 0.112 0.110 p = 1.0 0.078 0.025 0.081 0.071 0.068 0.125 0.101 0.085 0.109 0.110 p = 2.0 0.09…
Figure 11
Figure 11. Figure 11: Training distributions of Minkowski distances with r = 12. n = 1 n = 2 n = 3 n = 4 n = 5 n = 6 n = 7 n = 8 n = 9 n = 10 p = 0.5 0.056 0.016 0.083 0.057 0.067 0.120 0.098 0.097 0.109 0.114 p = 1.0 0.062 0.018 0.085 0.057 0.066 0.124 0.099 0.094 0.108 0.118 p = 2.0 0.07…
Figure 12
Figure 12. Figure 12: Training distributions of Minkowski distances with r = 14. n = 1 n = 2 n = 3 n = 4 n = 5 n = 6 n = 7 n = 8 n = 9 n = 10 p = 0.5 0.052 0.013 0.072 0.05 0.062 0.117 0.100 0.096 0.110 0.110 p = 1.0 0.056 0.015 0.074 0.05 0.062 0.121 0.101 0.092 0.108 0.113 p = 2.0 0.073 …
Figure 15
Figure 15. Figure 15: Model accuracy and hallucination F1 score vs r at different n for p = 0.5 r = 4 r = 6 r = 8 r = 10 r = 12 r = 14 r = 16 n = 1 0.61 0.62 0.66 0.65 0.62 0.63 0.63 n = 2 0.51 0.52 0.61 0.54 0.53 0.58 0.53 n = 3 0.49 0.46 0.49 0.46 0.48 0.51 0.48 n = 4 0.52 0.42 0.48 0.45…
Figure 16
Figure 16. Figure 16: Model accuracy and hallucination F1 score vs r at different n for p = 1.0 r = 4 r = 6 r = 8 r = 10 r = 12 r = 14 r = 16 n = 1 0.62 0.61 0.62 0.62 0.61 0.59 0.6 n = 2 0.48 0.54 0.55 0.55 0.54 0.53 0.55 n = 3 0.53 0.53 0.55 0.54 0.55 0.54 0.54 n = 4 0.41 0.44 0.51 0.48 …

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Forward citations

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Pith tools

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