A random covariance model for bi‐level graphical modeling with application to resting‐state fMRI data

We consider a novel problem, bi‐level graphical modeling, in which multiple individual graphical models can be considered as variants of a common group‐level graphical model and inference of both the group‐ and individual‐level graphical models is of interest. Such a problem arises from many applications, including multi‐subject neuro‐imaging and genomics data analysis. We propose a novel and efficient statistical method, the random covariance model, to learn the group‐ and individual‐level graphical models simultaneously. The proposed method can be nicely interpreted as a random covariance model that mimics the random effects model for mean structures in linear regression. It accounts for similarity between individual graphical models, identifies group‐level connections that are shared by individuals, and simultaneously infers multiple individual‐level networks. Compared to existing multiple graphical modeling methods that only focus on individual‐level graphical modeling, our model learns the group‐level structure underlying the multiple individual graphical models and enjoys computational efficiency that is particularly attractive for practical use. We further define a measure of degrees‐of‐freedom for the complexity of the model useful for model selection. We demonstrate the asymptotic properties of our method and show its finite‐sample performance through simulation studies. Finally, we apply the method to our motivating clinical data, a multi‐subject resting‐state functional magnetic resonance imaging dataset collected from participants diagnosed with schizophrenia, identifying both individual‐ and group‐level graphical models of functional connectivity.