Joint Hierarchical Bayesian Learning of Full-structure Noise for Brain Source Imaging

Many problems in human brain imaging involve hierarchical Bayesian (type-II maximum likelihood) regression models for observations with latent variables for source and noise, where parameters of priors for source and noise terms need to be estimated jointly from data. One example are biomagnetic inverse problems, where crucial factors influencing accuracy of brain source estimation are not only the noise level but also its correlation structure. Importantly, existing approaches have not addressed estimation of a full-structure noise covariance matrix. Using ideas from Riemannian geometry, we derive an efficient algorithm for updating both source and a full-structure noise covariance along the manifold of positive definite matrices. Our results demonstrate that the novel framework significantly improves upon state-of-the-art techniques in the real-world scenario with fully-structured noise covariance.