# Belief-Trajectory Energy: Measuring the Path to a Prediction

arXiv: https://arxiv.org/abs/2610.05114 · HTML: https://mlsanity.com/paper/2610.05114/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 27 of its 31 references that are in the map.
- First posted on arXiv 4 Oct 2026.

## Builds on (most cited first, 27 of 27)

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- [A Survey of Large Language Models](https://mlsanity.com/paper/2303.18223/) (2023)
- [Language Models (Mostly) Know What They Know](https://mlsanity.com/paper/2207.05221/) (2022)
- [OlympiadBench: A Challenging Benchmark for Promoting AGI with Olympiad-Level Bilingual Multimodal Scientific Problems](https://mlsanity.com/paper/2402.14008/) (2024)
- [OpenAI GPT-5 System Card](https://mlsanity.com/paper/2601.03267/) (2025)
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- [Release Strategies and the Social Impacts of Language Models](https://mlsanity.com/paper/1908.09203/) (2019)
- [DoLa: Decoding by Contrasting Layers Improves Factuality in Large Language Models](https://mlsanity.com/paper/2309.03883/) (2023)
- [Finding Neurons in a Haystack: Case Studies with Sparse Probing](https://mlsanity.com/paper/2305.01610/) (2023)
- [ProofWriter: Generating Implications, Proofs, and Abductive Statements over Natural Language](https://mlsanity.com/paper/2012.13048/) (2020)
- [Spotting LLMs With Binoculars: Zero-Shot Detection of Machine-Generated Text](https://mlsanity.com/paper/2401.12070/) (2024)
- [Patchscopes: A Unifying Framework for Inspecting Hidden Representations of Language Models](https://mlsanity.com/paper/2401.06102/) (2024)
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- [Toward Generalizable Evaluation in the LLM Era: A Survey Beyond Benchmarks](https://mlsanity.com/paper/2504.18838/) (2025)
- [Is Your Paper Being Reviewed by an LLM? Benchmarking AI Text Detection in Peer Review](https://mlsanity.com/paper/2502.19614/) (2025)
- [Disentangling Language and Culture for Evaluating Multilingual Large Language Models](https://mlsanity.com/paper/2505.24635/) (2025)
- [Model Utility Law: Evaluating LLMs beyond Performance through Mechanism Interpretable Metric](https://mlsanity.com/paper/2504.07440/) (2025)
- [EffiEval: Efficient and Generalizable Model Evaluation via Capability Coverage Maximization](https://mlsanity.com/paper/2508.09662/) (2025)
- [Beyond Benchmarks: Understanding Mixture-of-Experts Models through Internal Mechanisms](https://mlsanity.com/paper/2509.23933/) (2025)
- [SpecDetect: Simple, Fast, and Training-Free Detection of LLM-Generated Text via Spectral Analysis](https://mlsanity.com/paper/2508.11343/) (2025)

## Abstract (arXiv)

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/.