The Shape of Beliefs: Geometry, Dynamics, and Interventions along Representation Manifolds of Language Models' Posteriors
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arXiv
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| Autori principali: | , , , , , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866909015608393728 |
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| author | Sarfati, Raphaël Bigelow, Eric Wurgaft, Daniel Boppana, Siddharth Merullo, Jack Geiger, Atticus Lewis, Owen McGrath, Tom Lubana, Ekdeep Singh |
| author_facet | Sarfati, Raphaël Bigelow, Eric Wurgaft, Daniel Boppana, Siddharth Merullo, Jack Geiger, Atticus Lewis, Owen McGrath, Tom Lubana, Ekdeep Singh |
| contents | Large language models (LLMs) form implicit beliefs (posteriors over latent variables) from prompts, but we lack a mechanistic account of how these beliefs are encoded in representation space, how they update with new evidence, and how interventions reshape them. We study a controlled setting in which Llama-3.2 infers the parameters of a normal distribution from in-context samples. We show that parameter posteriors are encoded as curved manifolds in representation space, and trace how they evolve along the prompt. Standard linear steering moves representations off-manifold, inducing unintended, coupled changes, whereas geometry-aware methods preserve the target belief family. Our work demonstrates an example of linear field probing (LFP) as a principled approach to tile the data manifold and make interventions that respect the underlying geometry. Our results suggest that LLM beliefs are inherently geometric objects, and that globally linear representations are often inadequate abstractions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_02315 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Shape of Beliefs: Geometry, Dynamics, and Interventions along Representation Manifolds of Language Models' Posteriors Sarfati, Raphaël Bigelow, Eric Wurgaft, Daniel Boppana, Siddharth Merullo, Jack Geiger, Atticus Lewis, Owen McGrath, Tom Lubana, Ekdeep Singh Computation and Language Large language models (LLMs) form implicit beliefs (posteriors over latent variables) from prompts, but we lack a mechanistic account of how these beliefs are encoded in representation space, how they update with new evidence, and how interventions reshape them. We study a controlled setting in which Llama-3.2 infers the parameters of a normal distribution from in-context samples. We show that parameter posteriors are encoded as curved manifolds in representation space, and trace how they evolve along the prompt. Standard linear steering moves representations off-manifold, inducing unintended, coupled changes, whereas geometry-aware methods preserve the target belief family. Our work demonstrates an example of linear field probing (LFP) as a principled approach to tile the data manifold and make interventions that respect the underlying geometry. Our results suggest that LLM beliefs are inherently geometric objects, and that globally linear representations are often inadequate abstractions. |
| title | The Shape of Beliefs: Geometry, Dynamics, and Interventions along Representation Manifolds of Language Models' Posteriors |
| topic | Computation and Language |
| url | https://arxiv.org/abs/2602.02315 |