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We introduce Belief-Trajectory Energy (BTE), a novel model-grounded metric that quantifies an input by measuring the layerwise predictive revisions it induces within a Transformer. BTE maps intermediate model states into a shared predictive space, offering a principled characterization of belief change that can be summarized as a scalar or a depth profile. Empirically, BTE proves effective as a signal of task difficulty and supports fine-grained generator attribution and human-LLM review detection.
The internal journey of an LLM's prediction, not just its destination, reveals crucial information about input difficulty and generator identity.
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/.