Inverse solutions from MEEG are a direct reflect of the brain activity at a temporal resolution that no other in vivo noninvasive neuroimage could provide. Functional connectivity, per the definition of multivariate statistics in frequency domain of these inverse solutions unveil functional brain networks that strongly correlate with all cognition and behavior. An essential statistic is the cross-spectrum which represents all multivariate second-order properties of the functional brain networks and their connectivity. This cross-spectrum is determined as the Hermitian covariance matrix equivalent Estimating this cross-spectrum Simulations of such MEG or EEG inverse problem also reveal estimation errors of the functional connectivity determined by any of the state-of-the-art inverse solutions. We disclose a significant cause of estimation errors originating from misspecification of the functional network model incorporated into either inverse solution steps. We introduce the Bayesian identification of a Hidden Gaussian Graphical Spectral (HIGGS) model specifying such oscillatory brain networks model. In human EEG alpha rhythm simulations. Estimation errors measured as ROC performance do not surpass 2% in our HIGGS inverse solution and reach 20% in state-of-the-art methods. Macaque simultaneous EEG/ECoG recordings provide experimental confirmation for our inverse-solution with 1/3 more congruence according to Riemannian distances than state-of-the-art methods.