# Susceptibilities and Patterning: A Primer on Linear Response in Bayesian Learning

arXiv: https://arxiv.org/abs/2605.07980 · HTML: https://mlsanity.com/paper/2605.07980/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 33 references that are in the map.
- First posted on arXiv 8 May 2026.

## Builds on (most cited first, 12 of 12)

- [The Local Learning Coefficient: A Singularity-Aware Complexity Measure](https://mlsanity.com/paper/2308.12108/) (2023)
- [Covariances, Robustness, and Variational Bayes](https://mlsanity.com/paper/1709.02536/) (2017)
- [Loss Landscape Degeneracy and Stagewise Development in Transformers](https://mlsanity.com/paper/2402.02364/) (2024)
- [Differentiation and Specialization of Attention Heads via the Refined Local Learning Coefficient](https://mlsanity.com/paper/2410.02984/) (2024)
- [Bayesian Influence Functions for Hessian-Free Data Attribution](https://mlsanity.com/paper/2509.26544/) (2025)
- [Structural Inference: Interpreting Small Language Models with Susceptibilities](https://mlsanity.com/paper/2504.18274/) (2025)
- [The Loss Kernel: A Geometric Probe for Deep Learning Interpretability](https://mlsanity.com/paper/2509.26537/) (2025)
- [Patterning: The Dual of Interpretability](https://mlsanity.com/paper/2601.13548/) (2026)
- [Embryology of a Language Model](https://mlsanity.com/paper/2508.00331/) (2025)
- [Towards Spectroscopy: Susceptibility Clusters in Language Models](https://mlsanity.com/paper/2601.12703/) (2026)
- [W-Kernel and Its Principal Space for Frequentist Evaluation of Bayesian Estimators](https://mlsanity.com/paper/2311.13017/) (2023)
- [The Bayesian Infinitesimal Jackknife for Variance](https://mlsanity.com/paper/2305.06466/) (2023)

## Built on by (most cited first, 2 of 2)

- [Patterning in Practice: Debiasing Reward Models with Susceptibilities](https://mlsanity.com/paper/2609.00699/) (2026)
- [Interpretability for Turing Machines](https://mlsanity.com/paper/2609.04661/) (2026)

## Abstract (arXiv)

These notes introduce the theory of susceptibilities as developed in [arXiv:2504.18274, arXiv:2601.12703] for interpreting neural networks. The susceptibility of an observable φ to a data perturbation is defined as a derivative of a posterior expectation, which by the fluctuation--dissipation theorem equals a posterior covariance. Different choices of φ yield different objects: per-sample losses give the influence matrix (the Bayesian influence function of [arXiv:2509.26544]), while component-localized observables give the structural susceptibility matrix that pairs model components with data patterns. The susceptibility matrix is (up to a factor of nβ) the Jacobian of the map from data distributions to structural coordinates; its pseudo-inverse provides a linearized solution to the patterning problem of [arXiv:2601.13548]: finding data perturbations that produce a desired structural change. We motivate the theory from its statistical-mechanical foundations, then give a detailed exposition of susceptibilities, their empirical estimators, and their connection to the geometry of the loss landscape.