<p>Identifying the connectivity of functional networks underpinning undirectly observed phenomena for neurosciences or other fields poses a Bayesian inverse-problem. Electromagnetic (EEG or MEG) inverse-solutions unveil the cortical oscillatory networks that strongly correlate to brain function with a spectral transparency that no other in vivo neuroimage may provide. Simulations of such an inverse-problem also reveal distortions of the connectivity determined by most common state-of-the-art solutions. We disclose the origin of distortions and remedy them via a Hidden Gaussian Graphical Spectral (HIGGS) model, the Bayesian formalism for the inverse-problem in identifying such networks. In human EEG alpha rhythm simulations, distortions measured as ROC performance do not surpass the 2% in our HIGGS solution and reach 20% in state-of-the-art approaches. Congruence in macaque simultaneous EEG/ECoG recordings provides experimental confirmation for our solution with 1/3 more congruence than state-of-the-art methods.</p>