  This paper considers 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 are of interest. Such problem arises from
many applications including multi-subject neuroimaging 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 in
the group, and at the same time infers multiple individual-level networks.
Compared to existing multiple graphical modeling methods that only focus on
individual-level networks, our model learns the group-level structure
underlying the multiple individual networks 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 that can be used for
model selection. We demonstrate the asymptotic properties of the method and
show its finite sample performance through simulation studies. Finally, we
apply the proposed method to our motivating clinical data, a multi-subject
resting-state functional magnetic resonance imaging (fMRI) dataset collected
from schizophrenia patients.
