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

2026-05-08

AI Summary

This work investigates how to quantify the influence of data perturbations on observable quantities in Bayesian neural networks and establishes a mapping between data patterns and structural changes. Drawing on linear response theory, the authors define susceptibility via the posterior covariance and construct a susceptibility matrix that serves as the Jacobian of the map from data distribution to structural coordinates. The pseudoinverse of this matrix enables a linear solution to the inverse “patterning” problem, facilitating the design of data perturbations that induce desired structural modifications. By integrating Bayesian inference, the fluctuation–dissipation theorem, and geometric analysis of the loss landscape, this study unifies the theoretical formulations of influence functions and structural susceptibility, provides a computable framework for evaluating sensitivity, and offers an efficient linear approach to controlling neural network behavior.

Abstract

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.

Problem

• susceptibility
• Bayesian learning
• patterning problem
• data perturbation
• structural change

Innovation

• susceptibility
• Bayesian learning
• fluctuation-dissipation theorem
• influence function
• patterning