# Filtering Beats Fine Tuning: A Bayesian Kalman View of In Context Learning in LLMs

arXiv: https://arxiv.org/abs/2601.06100 · HTML: https://mlsanity.com/paper/2601.06100/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 19 references that are in the map.
- First posted on arXiv 2 Jan 2026.

## Builds on (most cited first, 5 of 5)

- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [LoRA: Low-Rank Adaptation of Large Language Models](https://mlsanity.com/paper/2106.09685/) (2021)
- [Optimizing Neural Networks with Kronecker-factored Approximate Curvature](https://mlsanity.com/paper/1503.05671/) (2015)
- [Transformers learn in-context by gradient descent](https://mlsanity.com/paper/2212.07677/) (2022)
- [Weight Uncertainty in Neural Networks](https://mlsanity.com/paper/1505.05424/) (2015)

## Abstract (arXiv)

We present a theory-first framework that interprets inference-time adaptation in large language models (LLMs) as online Bayesian state estimation. Rather than modeling rapid adaptation as implicit optimization or meta-learning, we formulate task- and context-specific learning as the sequential inference of a low-dimensional latent adaptation state governed by a linearized state-space model. Under Gaussian assumptions, adaptation follows a Kalman recursion with closed-form updates for both the posterior mean and covariance. This perspective elevates epistemic uncertainty to an explicit dynamical variable. We show that inference-time learning is driven by covariance collapse, i.e., rapid contraction of posterior uncertainty induced by informative tokens, which typically precedes convergence of the posterior mean. Using observability conditions on token-level Jacobians, we establish stability of the Bayesian filter, prove exponential covariance contraction rates, and derive mean-square error bounds. Gradient descent, natural-gradient methods, and meta-learning updates arise as singular, noise-free limits of the filtering dynamics, positioning optimization-based adaptation as a degenerate approximation of Bayesian inference. The resulting theory provides a unified probabilistic account of in-context learning, parameter-efficient adaptation, and test-time learning without parameter updates. It yields explicit guarantees on stability and sample efficiency, offers a principled interpretation of prompt informativeness via information accumulation, and clarifies the role of uncertainty dynamics absent from existing accounts. Minimal illustrative experiments corroborate the qualitative predictions of the theory.