Spectral Uncertainty Decomposition for Sparse Bayesian Learning

The paper introduces a spectral framework that decomposes the posterior covariance in sparse Bayesian learning for M/EEG source imaging into data‑informed and indeterminacy epistemic components, using the singular value structure of the prior‑preconditioned forward operator. It also presents an equivalent subtractive decomposition that expresses the prior covariance as the sum of posterior and data‑driven reduction terms, allowing precise accounting of uncertainty reduction through Bayesian learning. These complementary representations provide both theoretical insight and practical tools for interpretable uncertainty quantification in linear‑Gaussian Bayesian inverse solvers where the posterior covariance is available in closed form.