# Geometric Scaling of Bayesian Inference in LLMs

arXiv: https://arxiv.org/abs/2512.23752 · HTML: https://mlsanity.com/paper/2512.23752/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 17 references that are in the map.
- First posted on arXiv 27 Dec 2025.

## Builds on (most cited first, 8 of 8)

- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [Evaluating Large Language Models Trained on Code](https://mlsanity.com/paper/2107.03374/) (2021)
- [Layer Normalization](https://mlsanity.com/paper/1607.06450/) (2016)
- [Pythia: A Suite for Analyzing Large Language Models Across Training and Scaling](https://mlsanity.com/paper/2304.01373/) (2023)
- [What Does BERT Look At? An Analysis of BERT's Attention](https://mlsanity.com/paper/1906.04341/) (2019)
- [Analyzing Multi-Head Self-Attention: Specialized Heads Do the Heavy Lifting, the Rest Can Be Pruned](https://mlsanity.com/paper/1905.09418/) (2019)
- [Eliciting Latent Predictions from Transformers with the Tuned Lens](https://mlsanity.com/paper/2303.08112/) (2023)
- [Transformers represent belief state geometry in their residual stream](https://mlsanity.com/paper/2405.15943/) (2024)

## Built on by (most cited first, 3 of 3)

- [Mechanistic Tomography: Designed Measurement for Control-Oriented Interpretability](https://mlsanity.com/paper/2608.19338/) (2026)
- [Gradient Descent as Implicit EM in Distance-Based Neural Models](https://mlsanity.com/paper/2512.24780/) (2025)
- [Bayesian Wind Tunnels for Model Selection](https://mlsanity.com/paper/2607.19379/) (2026)

## Abstract (arXiv)

Recent work has shown that small transformers trained in controlled "wind-tunnel'' settings can implement exact Bayesian inference, and that their training dynamics produce a geometric substrate -- low-dimensional value manifolds and progressively orthogonal keys -- that encodes posterior structure. We investigate whether this geometric signature persists in production-grade language models. Across Pythia, Phi-2, Llama-3, and Mistral families, we find that last-layer value representations organize along a single dominant axis whose position strongly correlates with predictive entropy, and that domain-restricted prompts collapse this structure into the same low-dimensional manifolds observed in synthetic settings. To probe the role of this geometry, we perform targeted interventions on the entropy-aligned axis of Pythia-410M during in-context learning. Removing or perturbing this axis selectively disrupts the local uncertainty geometry, whereas matched random-axis interventions leave it intact. However, these single-layer manipulations do not produce proportionally specific degradation in Bayesian-like behavior, indicating that the geometry is a privileged readout of uncertainty rather than a singular computational bottleneck. Taken together, our results show that modern language models preserve the geometric substrate that enables Bayesian inference in wind tunnels, and organize their approximate Bayesian updates along this substrate.