Multi-Level Gaussian Graphical Models Conditional On Covariates

We address the problem of learning the structure of a high-dimensional Gaussian graphical model conditional on covariates, when each sample belongs to groups at multiple levels of hierarchy. The existing statistical methods for learning covariate-conditioned Gaussian graphical models focused on learning the aggregate behavior of inputs and outputs in a single-layer network. We propose a statistical model called multi-level conditional Gaussian graphical models for modeling multi-level output networks influenced by both individual-level and group-level inputs. We describe a decomposition of our model into a product of two components, one for sum variables and the other for difference variables derived from the original variables. This decomposition leads to an efficient learning algorithm for both complete data and incomplete data with randomly missing individual observations, as the expensive repeated computation of the partition function can be avoided. We demonstrate our method on simulated data and real-world data in finance and genomics.