Information Geometry meets Functional ANOVA: An Exact Fisher-Information Decomposition, with an Application to Radio Luminosity Functions

Abstract

Information geometry represents a fitted model as a manifold whose metric is the Fisher information. We show that, for a nonlinear regression model, this metric decomposes exactly across the Hoeffding-Sobol' (functional ANOVA) channels of its covariates. Treating the score contributions of main effects and interactions as vectors in a Hilbert space, their Gram matrix reproduces the (Gauss-Newton) Fisher information metric exactly, and this identity extends to an exact, additive decomposition of the Fisher information matrix into main-effect, interaction, and cross-channel terms: a Sobol'-type accounting for parameter information rather than output variance, and equivalently a channel-resolved pullback of the ambient Fisher metric along the model embedding. We give the exact decomposition theorem with a short proof (deriving both the centered and uncentered Fisher information), define information-theoretic analogues of Sobol' indices, characterize the sign and finite-sample behaviour of the cross-channel terms, and describe two practical estimators. As an astrophysical application we decompose the parameter information of a combined 1.4 GHz radio luminosity-function model spanning active galactic nuclei (AGN) and star-forming galaxies (SFG), fit to 6dFGS-NVSS and VLA-COSMOS 3 GHz data: the decomposition localizes where in luminosity each population and survey constrains the fit, and exposes the compensating ("sloppy") channel combinations near the luminosity-function turnover. A plant CO2-uptake experiment serves as a small, fully categorical companion example.