Advances in hierarchical Bayesian learning with applications to neuroimaging

Electro- and Magnetoencephalography (E/MEG) are non-invasive imaging techniques that enable us to measure brain electrical activity with high temporal resolution. This remarkable feature makes these imaging modalities indispensable tools in basic neuroscience and clinical neurology. The downside of both techniques, however, lies in their low spatial resolution. Brain source imaging (BSI) aims to mitigate this limitation, but is an inherently ill-posed inverse problem because of the large number of brain sources compared to the small number of sensors. To deal with the ill-posed characteristic of the BSI problem, various regularization techniques introducing prior assumptions on the spatio-temporal structure of the source activations have been proposed. One particularly successful approach is Type-II Bayesian learning, also called sparse Bayesian learning (SBL). The primary contribution of the first part of the thesis is to provide a unifying theoretical platform to reformulate three popular SBL algorithms for BSI under the majorization-minimization (MM) framework. This unification perspective allows us to compare different properties of SBL algorithms like their convergence behavior or reconstruction performance within a common theoretical framework. Building on the MM principle, we also propose a novel method that achieves favorable source reconstruction performance in low signal-to-noise-ratio settings. Precise knowledge of the parameters of the noise distribution, namely its scale and correlation structure, is a fundamental requirement for obtaining accurate solutions to many Type-II learning problems. In the second part of the thesis, we present novel principled techniques that can learn the noise covariance jointly with the distribution of the reconstructed sources within the framework of Type-II Bayesian learning. First, we derive analytic rules to update noise variance adaptively. Second, we propose spatial and temporal cross-validation schemes, where either subsets of E/MEG channels or recorded samples ...