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Estimating LLM Uncertainty with Evidence

T0 review · 4 major / 4 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read Probability-based confidence scores fail for LLMs because normalization erases the evidence strength that raw logits carry.

desk verdict A practically promising token-level uncertainty heuristic whose central theoretical claim is not supported by the proof as written; worth refereeing but not accepting without major revision. read the letter →

arxiv 2502.00290 v5 pith:H7IOBD4N submitted 2025-02-01 cs.CL cs.AI

Huan Ma , Jingdong Chen , Joey Tianyi Zhou , Guangyu Wang , Changqing Zhang This is my paper · ORCID
classification cs.CLcs.AI
keywords LLMuncertaintyestimationepistemicaleatoricDirichletevidencelogitshallucinationdetectiontokenreliabilitydynamicdecoding
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

Large language models often generate fluent answers to questions they know nothing about, and the usual probability-based confidence scores cannot flag this because softmax normalization erases how much evidence the model accumulated for each candidate during training. This paper argues that the raw logits before normalization retain that evidence, and introduces LogTokU, which treats the top $K$ logits as parameters of a Dirichlet distribution and derives two uncertainties for every generated token: aleatoric uncertainty (how torn the model is between candidates) and epistemic uncertainty (how much total evidence it has). The paper uses these signals to decide when a model should volunteer a second answer and to score whole responses for reliability, reporting higher AUROC than probability, entropy, and multi-sample methods on a question-answering benchmark and better multi-label accuracy in dynamic decoding. If the account holds, reliable token-level uncertainty is available in a single forward pass, with no sampling and no extra computation.

What carries the argument

The load-bearing machinery is the identification of raw top-$K$ logits with Dirichlet evidence parameters, borrowed from evidential learning. Here $\alpha_k$ is the $k$-th largest pre-softmax score and $\alpha_0=\sum_{k=1}^K \alpha_k$ is the total evidence; aleatoric uncertainty is the expected entropy of the resulting Dirichlet, and epistemic uncertainty is $K/(\alpha_0+K)$, so it shrinks as the total logit strength grows. This machinery carries the argument because $\mathrm{EU}$ depends on logit magnitude rather than normalized probability, which is exactly the information the paper claims softmax destroys. The paper supports the identification with a gradient analysis that treats cross-entropy training as an evidential-learning loss plus an evidence-regularization term.

What would settle it

Run any real LLM generation and inspect tokens where all top-$K$ logits are negative: Eq. 5 returns a negative epistemic uncertainty, which the Dirichlet interpretation cannot accommodate. Alternatively, fine-tune on samples in which the correct token is not among the top $K$ predictions, for which the paper's Theorem 1 predicts the sum of top-$K$ logits strictly increases while the gradient update actually decreases it; measuring the summed top-$K$ logits over training would settle the claim.

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Extended reading notes

Core claim

The paper's central claim is that probability-based reliability estimation fails because normalization discards the strength of evidence accumulation, and that this information can be recovered from raw logits. Concretely, it sets $\alpha_k = M(\tau_k)$ for the $k$-th largest logit, takes the Dirichlet distribution over these evidence parameters, and defines token uncertainty as $\mathrm{AU}(a_t) = -\sum_{k=1}^K \frac{\alpha_k}{\alpha_0}(\psi(\alpha_k+1)-\psi(\alpha_0+1))$ and $\mathrm{EU}(a_t) = K/\sum_{k=1}^K (\alpha_k+1)$. High total evidence spread over several candidates means the model knows more than one valid answer; low total evidence with one dominant candidate means it lacks knowledge but is making a suggestion. This decoupling separates 'I do not know' from 'I know more than one answer', the two cases that probability conflates, and the paper demonstrates the separation through a four-quadrant taxonomy of token states and two downstream uses.

Load-bearing premise

The whole construction rests on treating the sizes of the raw pre-softmax scores as a faithful measure of how much similar evidence the model saw during training; if logit magnitude is not a reliable evidence counter, the uncertainty formula and both downstream uses lose their foundation.

Editorial extensions

If this is right

  • Token-level uncertainty is available in a single forward pass, so hallucination warnings and reliability scores can be computed in real time without sampling.
  • Dynamic decoding can use epistemic uncertainty to keep sampling diverse when the model knows several answers and to sample cautiously when knowledge is missing, improving multi-label accuracy across several model sizes.
  • Response reliability can be read from the least reliable tokens, removing the need to hand-assign weights to uncritical tokens such as commas and function words.
  • The method does not apply to black-box APIs or distilled models, because those settings either do not expose logits or rescale them so that evidence strength is lost.

Reading between the lines

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

  • Editorial inference: if logit magnitude is a faithful evidence counter, then post-hoc interventions that shift logit scales, such as temperature scaling, calibration layers, or distillation, should degrade LogTokU's uncertainty even when they leave accuracy unchanged; that is a direct, testable consequence the paper does not run.
  • Editorial inference: the four-quadrant taxonomy suggests that tokens in Quadrant IV (high AU, low EU) should be treated as safe rather than uncertain in hallucination detectors, and comparing $\mathrm{AU}\cdot\mathrm{EU}$ scoring with $\mathrm{EU}$-only scoring on free-form QA would isolate whether the typology earns its keep.
  • Editorial inference: the same evidence readout could drive abstention or retrieval decisions during generation, deciding when to look up external knowledge, rather than only rescoring text after it is produced.
  • Editorial inference: because Eq. 5 uses raw logits directly, a constant added to all logits changes EU without changing the output distribution; probing this sensitivity would test whether the evidence interpretation is intrinsic to the model or an artifact of logit scale.
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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 / 4 minor

Summary. The paper argues that probability-based token-level uncertainty in LLMs fails because softmax normalization discards the evidence-strength information carried by raw logits. It proposes LogTokU, which interprets the top-K logits as Dirichlet evidence parameters and computes two uncertainty components (aleatoric and epistemic) in a single forward pass. The estimated uncertainty is applied to dynamic decoding on SemEval and to response-reliability estimation (AUROC against correctness) on TruthfulQA, across LLaMA-2 and LLaMA-3 models of several sizes. The theoretical justification is presented in Section 3.5 and Appendix G, where a theorem claims that cross-entropy training accumulates top-K evidence.

Significance. If the central claim held, LogTokU would be a practically attractive contribution: it gives token-level uncertainty without sampling, distinguishes "I do not know" from "I know more than one answer," and supports downstream hallucination-related applications. The four-quadrant framing is intuitive, and the experiments cover a useful range of model sizes with several sampling-based baselines. The appendix also provides implementation details for the baselines and an honest limitations section. However, the core theoretical result is not established: the proof in Appendix G concerns direct logit updates rather than trained network parameters, and the Dirichlet parameterization in Eq. (3) is invalid for ordinary negative logits. The empirical gains are suggestive but currently best interpreted as evidence for a heuristic; the reported AUROC and decoding improvements also lack error bars and rely on parameters selected on the evaluation data.

major comments (4)
  1. [Appendix G, Theorem 1 and Eq. (16)] The proof analyzes gradient descent on the logits themselves (update rules z_k := z_k + eta(1-p_k) and z_i := z_i - eta p_i), not on the network parameters of M, so the conclusion does not apply to an LLM trained by optimizing weights. It also assumes the correct class tau_k lies in the top-K set T; for a misclassified sample with tau_k not in T, the change in the top-K sum is -eta sum_{i in T} p_i < 0, directly contradicting the claimed accumulation. Because top-K membership can change across gradient steps, the fixed-set inequality in Eq. (16) cannot establish the theorem as stated.
  2. [Eq. (3), Eq. (5), and Section 3.5] Setting alpha_k = M(tau_k) to raw logits violates the requirement that Dirichlet parameters be strictly positive, since real LLM top-K logits are frequently negative. When sum_k (alpha_k + 1) <= 0, the epistemic uncertainty EU = K / sum_k(alpha_k + 1) is undefined or negative. The "Bridging the positivity gap" paragraph in Section 3.5 argues via the softmax exponential that negative logits behave like weak evidence, but Eq. (3) still uses raw logits rather than exponentials or ReLU, so the implementation is run precisely in a regime where the stated Dirichlet justification fails.
  3. [Tables 1-3 and Figure 4] The headline results are reported without error bars or significance tests, and the main free parameters are selected on the evaluation data. Appendix C.1 chooses K by best AUROC on LLaMA2-13B, and Figure 4 selects the decoding threshold at the best accumulated-score point on the same test set; no validation split or sensitivity analysis is provided. As a consequence, the magnitude of the claimed improvements over the baselines is not established by the current experimental protocol.
  4. [Appendix F vs. Theorem 1] Appendix F states that distilled models cannot be handled because they "lose the strength of evidence in their logits," but Theorem 1 claims evidence accumulation for any LLM trained with cross-entropy. Distillation is typically trained with distribution-matching losses that are cross-entropy-like, so the scope of the theoretical claim needs to be reconciled with this self-reported limitation, or the theorem must be restricted accordingly.
minor comments (4)
  1. [Abstract] The abstract says the code is available at "link"; this placeholder should either contain a working repository or be removed before publication.
  2. [Eq. (3) and Eq. (7)] The symbol K is used both for the number of top logits in Eq. (3) and for the number of lowest-reliability tokens aggregated in Eq. (7); using distinct symbols would remove ambiguity.
  3. [Eq. (8) and Appendix A] Equation (8) defines R(at) = -AU(at) * EU(at), while Appendix A defines unrel(word) = AU(word) * EU(word) with lower values indicating unreliability; the sign convention should be stated consistently in both places.
  4. [Appendix C.2] The first sentence of Appendix C.2 misspells BLEURT as "BLUERT".

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: LogTokU's uncertainty is defined from logits and validated against external correctness labels; Appendix G's proof flaw is a validity issue, not a circular reduction.

full rationale

I walked the derivation chain. Eq. (3) sets alpha_k = logits, and Eq. (5) defines EU as K / sum(alpha_k + 1); these are definitions, not predictions. The paper's claim that probability normalization loses evidence strength is a premise, supported by the softmax invariance and illustrative examples, and the empirical validation (Tables 1 and 2) uses external correctness labels from SemEval and TruthfulQA, so the reported improvements are not forced by the construction of EU. The theoretical proof in Appendix G is not circular, but it is formally deficient: it assumes the correct class is already in the current top-K set ('Let T represent the set of classes corresponding to the top predicted classes (tau_k in T)'), and it analyzes the update of the old top-K set rather than guaranteeing the update of the new top-K set after the gradient step; Eq. (3) also does not enforce the nonnegativity that Dirichlet parameters require. These are correctness and validity concerns, not circularity. The benchmark-specific choice of K in Appendix C.1 appears to be test-set tuning, which weakens the reported AUROC as an unbiased estimate but does not make the prediction equivalent to the fitted value. No load-bearing self-citations were found; reference [22] is an external prior work. Therefore, no circular step can be exhibited under the required standard.

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

The central method introduces one structural choice K and a decision threshold; both are tuned on the evaluation data. The theoretical foundation rests on the unproven premise that raw logit magnitudes track training evidence and on an invalid gradient analysis. No new physical entities are introduced.

free parameters (3)
  • K (number of top logits used as evidence) = 20 (best in Table 3 ablation; not fixed in main text)
    K controls alpha_0 and therefore EU/AU; Table 3 shows AUROC swings from 65.7% at K=1 to 79.8% at K=20 and back to 69.1% at K=all, so the reported results depend on picking K on the evaluation benchmark.
  • EU threshold for dynamic decoding = not reported
    Section 4.2.1 says the LLM selects one class when EU is high and two when low, but never states how the threshold is chosen; if tuned on SemEval test data, the reported gains include fitting.
  • K in response reliability aggregation (Eq. 7) = not clearly separated from top-logit K; likely 20
    R_response averages the K tokens with lowest reliability; the same symbol K is used for the top-logit window and the number of tokens averaged, and the ablation in Table 3 tunes it on the evaluation set.
assumptions (4)
  • ad hoc to paper Logit magnitudes accumulate evidence during training proportional to how often similar contexts and answers were seen.
    Introduced in Sec. 3.2 and Fig. 1(b); Appendix G attempts a proof but the proof assumes logits are directly optimized.
  • ad hoc to paper Raw top-K logits are valid Dirichlet distribution parameters alpha_k.
    Eq. 3 sets alpha_k = M(tau_k). Dirichlet parameters must be nonnegative, but LLM logits can be negative; the paper mentions ReLU only as a naive option and does not apply it.
  • ad hoc to paper Gradient descent on cross-entropy can be analyzed as if the optimized variables are the logits themselves.
    Appendix G Theorem 1 computes partial derivatives with respect to logits and updates logits directly, which does not describe training of the transformer parameters that produce logits.
  • domain assumption BLEURT > 0.5 (and the LLM-judge variant) correctly labels TruthfulQA responses as truthful or false.
    Section 5.2.1 uses BLEURT > 0.5 as the correctness label for AUROC; this is a standard but noisy proxy for truthfulness.

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Cite this review

Pith. "Pith review of Estimating LLM Uncertainty with Evidence." pith.science (2026). https://pith.science/paper/H7IOBD4N

@misc{pith2026250200290,
  author       = {Pith},
  title        = {Pith review of: Estimating LLM Uncertainty with Evidence},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/H7IOBD4N}},
  note         = {Machine review of arXiv:2502.00290}
}
read the original abstract

Over the past few years, Large Language Models (LLMs) have developed rapidly and are widely applied in various domains. However, LLMs face the issue of hallucinations, generating responses that may be unreliable when the models lack relevant knowledge. To be aware of potential hallucinations, uncertainty estimation methods have been introduced, and most of them have confirmed that reliability lies in critical tokens. However, probability-based methods perform poorly in identifying token reliability, limiting their practical utility. In this paper, we reveal that the probability-based method fails to estimate token reliability due to the loss of evidence strength information which is accumulated in the training stage. Therefore, we present Logits-induced token uncertainty (LogTokU), a framework for estimating decoupled token uncertainty in LLMs, enabling real-time uncertainty estimation without requiring multiple sampling processes. We employ evidence modeling to implement LogTokU and use the estimated uncertainty to guide downstream tasks. The experimental results demonstrate that LogTokU has significant effectiveness and promise.

Figures

Figures reproduced from arXiv: 2502.00290 by the authors.

Figure 1
Figure 1. Why probability-based methods fail? Left: A pair of examples on LLaMA-2 shows that probability fails in estimating reliability. Since LLMs know the names of many presidents, the probability after normalization is very low; whereas for the future of the universe, since LLMs only know one hypothesis, the probability is very high. The probability-based reliability measure is counterintuitive, as the answers on common s… view at source ↗
Figure 2
Figure 2. Why LogTokU works? Left: Illustration of four different scenarios considered in LogTokU, where the gray bars represent the logits for predicting the next token, the triangular patterns represent the corresponding Dirichlet distribution, and the table below compares uncertainty estimation using probability with that using LogTokU. Right: A case study from a medical QA, where the markings under each word reflect relia… view at source ↗
Figure 3
Figure 3. Illustration of experimental setting in Table 1. [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: A close-up observation explains why LogTokU achieves the best [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

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    Token id: pos 1066, itive 3321, please note that the word “positive” referred here is different from another word “ positive”

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    Gradient for incorrect classes ( τ i ̸= τ k): ∂L ∂zτ i = pτ i . These gradients describe how the loss changes with respect to the logits. Using gradient descent with a learning rate η, the update rules for the logits are as follows:

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    These updates ensure that the logit for the correct class increases, while the logits for incorrect classes decrease

    Update for incorrect classes ( τ i ̸= τ k): zτ i := zτ i − ηpτ i . These updates ensure that the logit for the correct class increases, while the logits for incorrect classes decrease. Next, we analyze the change in the total evidence, defined as the sum of the k-largest logit...

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    Available: https://arxiv.org/abs/2305.19187

    [Online]. Available: https://arxiv.org/abs/2305.19187

    arXiv

Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.