The Shape of Beliefs: Geometry, Dynamics, and Interventions along Representation Manifolds of Language Models' Posteriors

Large language models (LLMs) represent prompt-conditioned beliefs (posteriors over answers and claims), 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 generates samples from a normal distribution by implicitly inferring its parameters (mean and standard deviation) given only samples from the distribution in context. We find representations of curved "belief manifolds" for these parameters form with sufficient in-context learning and study how the model adapts when the distribution suddenly changes. While standard linear steering often pushes the model off-manifold and induces coupled, out-of-distribution shifts, geometry and field-aware steering better preserves the intended belief family. Our work demonstrates an example of linear field probing (LFP) as a simple approach to tile the data manifold and make interventions that respect the underlying geometry. We conclude that rich structure emerges naturally in LLMs and that purely linear concept representations are often an inadequate abstraction.