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Belief-Trajectory Energy: Measuring the Path to a Prediction

Jiahao Ying, Wei Tang, Boxian Ai, Yaoning Wang, Haotian Chen, Wenhe Sun, Caijun Xu, Haozhan Cai, Changyi Xiao, Yixin Cao

arXiv:2610.05114v1cs.CL

TL;DR

LLM evaluation typically sees only final predictions, leaving layerwise predictive revisions unused. BTE measures those revisions in a shared predictive space and preserves them as a scalar or depth profile. Across the reported applications, it provides a model-relative difficulty signal and supports review detection and generator attribution.

  • Problem

    LLM evaluation typically observes final outputs, leaving the trajectory of internal predictive change largely unused despite its potential to distinguish inputs beyond their final outputs.

    Source: §1 · pp. 1–2

  • Method

    BTE reads intermediate states as predictive distributions and measures their successive revisions, aggregating them as a scalar or depth profile.

    Source: §2 · pp. 2–3, §1 · p. 2

  • Results

    BTE provides a model-relative difficulty signal across reasoning tasks and supports review detection and generator attribution, with 0.998 macro-averaged AUROC and 95.6% eight-way source-attribution accuracy reported.

    Source: §1 · p. 2

  • Takeaways & Limitations

    BTE offers a complementary model-grounded view of data, and its measurement reflects the scorer’s perspective and learned training exposure.

    Source: §4 · p. 10, §7 · p. 11, §6 · p. 11

    BTE is not expected to reproduce human-defined properties exactly, because its measurement can vary with the scoring model and the data being scored.

    Source: §6 · p. 11

Abstract

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Large language models (LLMs) progressively revise their predictions across Transformer layers, yet we typically observe only the final output, discarding the trajectory through which it is formed. We introduce Belief-Trajectory Energy(BTE), a model-grounded measure that characterizes an input through the layerwise predictive revisions it induces in a model. By mapping intermediate states into a shared predictive space, BTE provides a principled measure of belief change that can be summarized as either a scalar or a structured depth profile. Theoretically, we show that local BTE corresponds to predictive revision under the Fisher-Rao geometry, while the sequence of revisions captures information beyond the initial-to-final belief change. Empirically, scalar BTE provides a model-relative signal of difficulty across diverse reasoning tasks, while richer BTE representations support human-LLM review detection and fine-grained generator attribution, reaching up to $0.998$ macro-AUROC and $95.6\%$ eight-way attribution accuracy. Further analysis shows that BTE develops throughout pretraining and is selectively reshaped by targeted training, demonstrating that the resulting measurement reflects what the scoring model has learned. Together, our results establish belief trajectories as a principled model-grounded signal and suggest a broader perspective in which learned models can themselves serve as instruments for characterizing the data they process. More demonstrations can be found at https://yingjiahao14.github.io/BTE-web/.

1 Introduction

LLM evaluation usually observes only final outputs, leaving the internal trajectory of predictive change unused. The paper introduces BTE to characterize inputs through the predictive revisions they induce in a model.

1 Introduction

  • A useful trajectory measure must be principled, compact, comparable, and applicable across tasks without relying on task-specific supervised predictors.The challenge is to preserve evolution across layers at an intermediate granularity between neuron-level analysis and endpoint statistics.Source: §1 · p. 2
  • BTE reads intermediate states as next-token predictive distributions and measures successive revisions in this shared predictive space.Aggregating revisions across tokens yields a depth profile that can also be reduced to a scalar.Source: §1 · p. 2
  • Locally, BTE corresponds to predictive revision under Fisher–Rao geometry, while its sequence across depth preserves information unavailable from the endpoint change alone.The trajectory therefore retains intermediate revisions that cannot be recovered from the initial-to-final belief change.Source: §1 · p. 2
  • BTE provides a training-free, model-relative difficulty signal and supports human–LLM review detection and generator attribution, reaching 0.998 macro-averaged AUROC and 95.6% eight-way source-attribution accuracy.Difficulty orderings can differ across scoring models; the review result is reported for paper-disjoint detection across eight conference-years.Source: §1 · p. 2

2 The BTE Statistic

BTE maps intermediate model states into predictive distributions and measures how beliefs revise across layers. Token-level revisions form trajectories that can be interpreted locally, preserved across depth, and aggregated into a fixed-length profile.

2 The BTE Statistic

  • Intermediate states are read through the model’s output head as predictive distributions, whose successive revisions define each token’s belief trajectory and BTE.The model’s vocabulary projection provides a shared predictive space for comparing states across layers.Source: §2 · pp. 2–3

2.1 Interpreting Belief Trajectory Energy

  • A local BTE step measures effective predictive revision: differential logit updates among probable tokens matter, while a uniform shift contributes nothing.Locally, Jensen–Shannon divergence corresponds to Fisher–Rao geometry in logit coordinates and has an information-theoretic interpretation in nats.Source: §2.1 · pp. 3–4
  • The endpoint displacement is only a lower bound on cumulative revisions; distinct intermediate trajectories can yield different BTE sequences even with identical endpoint beliefs.Unlike likelihood or perplexity, which depend on the final predictive distribution, the trajectory retains the path through which that prediction forms.Source: §2.1 · p. 4

2.2 BTE Depth Profile

  • Mean-pooling token-level BTE at each depth produces a fixed-length profile that retains depth structure while normalizing out response length.Each profile entry is the average predictive revision per token at that depth; the local values are bounded by log 2, and the top 5% of tokens account for only about 8% of observed response differences.Source: §2.2 · p. 4

3 Experimental Analysis

The experiments test BTE as a training-free, model-relative difficulty signal and use its depth profile for human–LLM review detection and generator attribution. Results show useful signals across these tasks, with performance depending on the scorer and representation.

3 Experimental Analysis

  • The evaluation moves from a scalar central-layer difficulty readout to depth-profile review detection and token-level generator attribution, using Llama-3.1-8B-Instruct by default as scorer.The difficulty analysis compares reference-solution and problem-only BTE, and varies the scorer; review detection tests shape-only and full profiles across three training settings.Source: §3 · pp. 4–5, §3.1 · p. 5, §3.2 · p. 7

3.1 BTE as a Difficulty Indicator

  • Response BTE increases with difficulty on all four within-task benchmarks (ρ=0.33–0.66), but cross-task ordering varies by scorer and problem-only BTE can miss reasoning demands.On ProofWriter and SpaRTUN, response BTE correlations are 0.33 and 0.66, versus problem-only correlations of −0.01 and 0.08; difficulty sensitivity also depends on scorer capability and familiarity.Source: §3.1 · pp. 5–6, Figure 3 · p. 6

3.2 BTE as a Review Detector

  • BTE profiles generalize across papers and held-out generator families, while newer generators’ profiles approach or exceed human profiles as their human-evaluated review preference rises.Under cross-family transfer, Qwen-BTE reaches 0.995 macro-AUROC and 0.994 after rewriting; the signed middle-layer BTE shift correlates with human preference across generators (ρ=0.857, p=0.024).Source: §3.2 · pp. 7–8, Figure 4 · p. 8
  • 0.998 macro-AUROC for Qwen-BTE in full-coverage review detection exceeds 0.901 for Open-Anchor and the below-0.868 likelihood-based baselines.Performance remains nearly unchanged after rewriting human reviews.Source: Table 1 · p. 7, §3.2 · p. 8
  • Scorer familiarity can weaken difficulty comparability across data distributions yet strengthen review discrimination when familiarity differs systematically between human and generated reviews.The proposed explanation for Qwen’s greater familiarity with generated text is tentative because training mixtures are not public.Source: §3.2 · pp. 8–9

3.3 BTE for Generator Attribution

  • Token-level BTE variation raises eight-way attribution accuracy from 86.4% to 93.2% with Llama and from 91.3% to 95.6% with Qwen.The remaining confusions are concentrated between GPT-5.5 and GPT-5.6.Source: §3.3 · p. 9, Table 2 · p. 9

4 Ablation Study

The analysis examines how BTE develops during pretraining and changes with targeted fine-tuning, using OLMo-2-7B checkpoints and mathematical-reasoning data. The results connect these changes to the scoring model’s training and familiarity with the scored distribution.

Study design

  • The pretraining analysis traces OLMo-2-7B checkpoints, while the targeted-exposure analysis fine-tunes Llama-3.1-8B-Instruct on human-verified PRM800K mathematical solutions.The latter uses mathematical reasoning to provide controlled difficulty labels and data distributions with differing model exposure.Source: §4 · p. 9

Pretraining

  • Middle-layer BTE rises seven-fold from 0.013 to 0.093 as OLMo-2-7B pretraining progresses, while its correlation with MATH difficulty shifts from ρ= −0.13 at 1T tokens to ρ= +0.03 at 4T tokens.The randomly initialized model’s five MATH difficulty levels have identical middle-layer means of 0.013 and standard deviation below 0.0002 across 500 solutions.Source: §4 · pp. 9–10

Fine-tuning and targeted exposure

  • After MATH fine-tuning, middle-layer BTE decreases 22% on MATH-500 versus 17% on GSM8K; within MATH, fine-tuning on levels 4–5 reduces level-5 BTE more than level-1 (−19.7% vs. −17.5%).Source: §4 · p. 10

Interpretation

  • BTE’s structure develops during broad pretraining and shifts with targeted exposure, making the scorer’s training history part of this model-relative measurement.The authors describe targeted exposure as associated with smaller predictive revisions on the exposed distribution.Source: §4 · p. 10

5 Related Work

Prior work examines model internals and uses intermediate states to infer properties of inputs or outputs. BTE instead measures successive predictive changes and offers both a training-free scalar and a depth-wise representation.

Related work

  • BTE shifts attention from interpreting individual intermediate states to measuring their successive changes in predictive-distribution space.Prior approaches often use high-dimensional representations or task-specific supervised readouts, whereas BTE provides a training-free scalar and compact depth-wise representation.Source: §5 · p. 10

6 Discussion

BTE measures data from a model’s learned perspective, so it need not match human-defined properties exactly. Its behavior depends on both the scoring model and what is being scored, and the authors identify representation and measurement as open questions.

Discussion

  • BTE is a complementary, model-grounded view of data, not a measure expected to reproduce human-defined properties exactly.Difficulty measurements can vary with the scoring model and the scored input; how internal computation should be represented and measured remains open.Source: §6 · p. 11

7 Conclusion

The paper concludes that BTE measures the predictive revisions an input induces across Transformer layers and provides useful scalar and structured signals. Its results support treating learned models as instruments for characterizing data through their internal dynamics.

Conclusion

  • BTE characterizes inputs through predictive revisions across Transformer layers and provides a scalar signal and structured representation useful across several tasks.The reported applications include difficulty measurement, human–LLM review detection, and generator attribution.Source: §7 · p. 11

A.1 Additional Theoretical Analysis

The analysis expands around ε = 0, where the first-order term vanishes, and applies a first-order softmax expansion to prove the result.

  • The midpoint distribution is defined as M = (p + q)/2 = p + ε/2 before expansion around ε = 0.Source: §A.1 · p. 14
  • The first-order term vanishes because the components of ε sum to zero.Source: §A.1 · p. 14
  • For p = softmax(z) and q = softmax(z + δ), the proof uses a first-order expansion of the softmax.Source: §A.1 · p. 14

A.2 Difficulty Experiments: Data and Readout Windows

The difficulty experiments compare fixed scalar summaries of BTE across benchmarks and check whether solution-writing style affects the observed mathematical-task trend.

A.2 Difficulty Experiments: Data and Readout Windows

  • Difficulty benchmarks use official solutions as scorer input and compute BTE on solution tokens; ProofWriter theories and SpaRTUN stories define bootstrap clusters.Source: §A.2 · p. 14, Table 3 · p. 15
  • The middle-layer readout is the only fixed scalar readout positively associated with difficulty across all five settings, with 95% confidence intervals excluding zero.The full-depth mean performs best in the cross-benchmark mathematical comparison, while the middle window has the most consistent direction across tasks.Source: §A.2 · p. 14, Table 4 · p. 15
  • After GPT-5.5 rewrites all 150 mathematical benchmark solutions under one instruction, the GSM8K < MATH-500 < OlympiadBench ordering remains and the middle-layer trend strengthens (ρ = 0.44 → 0.75).Source: §A.2 · p. 15, Table 5 · p. 15

A.3 Review Data and Detector Details

The review experiments evaluate BTE representations and reference detectors across paper-disjoint data splits, scorer variants, and test conference-years. They report year-wise variation and a generated-versus-human review predictability gap.

A.3 Review Data and Detector Details

  • The corpus covers 11,200 review pairs from seven generators across eight conference-years, with paper-disjoint splits; the common test set has 1,033 pairs per generator.Rewritten reviews are used only for evaluation, never classifier fitting.Source: §A.3 · pp. 15–16, Table 6 · p. 16
  • The full BTE depth profile is the main representation; depth-wise normalization removes review-level magnitude and scale while retaining the relative pattern of revisions.Source: §A.3 · pp. 15–16
  • Settings A and B train on ICLR 2021 only, whereas C includes all seven generators and eight conference-years and more training data, so their differences do not isolate one factor.Source: §A.3 · p. 16
  • Reference detectors include log-likelihood, SpecDetect, Binoculars, and a transductive Open-Anchor variant, with Open-Anchor evaluated on 713 rather than 1,033 pairs per generator.Its anchor-source reviews are excluded, so its test set is not identical to the other detectors'.Source: §A.3 · p. 16
  • Generated reviews are easier to predict than human reviews for every scorer; the gap grows across Qwen generations (5.0%, 6.4%, 10.4%) and is 2.4% for Llama.Source: §A.3 · pp. 16–17, Table 8 · p. 17
  • Across test years, NeurIPS 2024 has the minimum macro-AUROC in every setting and for both scorers; the 2024 cells may be an upper bound on that year's difficulty because human reviews may have LLM assistance.Exposure to more generators and years narrows the year-to-year spread, from 0.18 to 0.05 for Llama and to 0.007 for Qwen.Source: §A.3 · p. 17

A.4 Human Evaluation

The human-review study finds that several newer generators are preferred over matched human reviews, and that BTE shifts track those preferences. The section also examines attribution features and shows that response length qualifies the interpretation of BTE as a difficulty signal.

A.4 Human Evaluation

  • The study compares human and generated reviews of 30 papers, using two blinded evaluators per pair and a nine-dimension rubric with Overall as the primary outcome.It collected 240 reviews and 600 judgments; the Overall score assigns 1 to a model win, 0.5 to a tie, and 0 to a human win.Source: §A.4 · p. 17, A B C · pp. 19–20

A B C

  • GPT-5.5, GPT-5.6, and Claude Opus 5 score 0.867, 0.892, and 0.875, respectively; their pooled score is 0.878 versus 0.531 for three legacy generators.Gemini 3.1 is an exception among newer generators, with a preference score of 0.483.Source: A B C · p. 20
  • Across seven generators, the signed middle-layer BTE shift correlates with Overall preference at Spearman’s ρ=0.857.The association is similar for Llama (ρ=0.786, p=0.048) and Qwen (ρ=0.857, p=0.024).Source: A B C · p. 21

A.5 Attribution: Feature Sets and Transfer Settings

  • Token-level BTE summaries form 32D–352D attribution features; richer token statistics separate Human and legacy-generator reviews from newer ones, while GPT-5.5 and GPT-5.6 overlap.The feature sets summarize layerwise token distributions, position bins, autocorrelation, and high-energy tokens.Source: §A.5 · p. 21, Figure 9 · p. 22

A.6 Response-Length Sensitivity

  • Length alone is a weak review detector, and removing its fitted linear component changes residual-profile AUROC by at most 0.008.The reported length-only macro-AUROC range is 0.512–0.538.Source: §A.6 · p. 23
  • Controlling for response length attenuates BTE’s association with difficulty, so BTE is not a length-independent difficulty measure.After controlling for response and problem length, a residual association remains for Llama and OpenMath2, but not for Nemotron.Source: §A.6 · p. 23, Table 16 · p. 23

A.7 Fine-tuning details

  • The appendix specifies PRM800K solution data, one-epoch Llama fine-tuning, and OLMo-2-7B checkpoints used to examine BTE across training.The training setup includes checkpoints at 1,000, 5,000, and 9,704 solutions; the checkpoint comparison includes earlier pretraining stages.Source: §A.7 · p. 24

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