Exact Spectral Uncertainty Decomposition for Linear-Gaussian Bayesian M/EEG Inverse Problems

Abstract

Distributed magneto- and electroencephalographic (M/EEG) source reconstruction is a severely underdetermined linear inverse problem. Bayesian source imaging regularizes inversion through source-prior and sensor-noise covariances, but the posterior covariance does not directly distinguish uncertainty constrained by the measurements from uncertainty unobservable under the sensing geometry. We study this distinction for linear-Gaussian M/EEG models with diagonal source-prior and sensor-noise covariances, covering Type-I inference and Type-II inference conditional on learned covariance hyperparameters. After restricting the analysis to the prior-supported source space, standardizing the prior, and whitening the sensor noise, the complete singular spectrum of the forward operator separates likelihood-visible and likelihood-null directions. This yields an additive uncertainty decomposition of the conditional posterior covariance: uncertainty in likelihood-visible directions is reduced according to spectral information content, whereas likelihood-null uncertainty remains unchanged from the prior. The spectral representation also yields a prior-minus-reduction identity, showing that covariance reduction is confined to the likelihood-visible subspace. Spectral markers quantify likelihood information and prior-variance retention and reduction. Fixed-orientation MEG simulations with adaptive noise learning illustrate the framework. Dense Bayesian minimum-norm inference retains a large likelihood-null component, whereas sparse $\gamma$-MAP strongly reduces the fitted prior support. Within that support, the markers distinguish strongly informed directions from a weakly constrained direction. Thus, prior-support selection determines which source coordinates remain in the fitted model, while the spectral decomposition quantifies how strongly the retained directions are constrained by the measurements.