Inferring two-level hierarchical Gaussian Graphical Models to discover shared and context-specific conditional dependencies from high-dimensional heterogeneous data

Gaussian graphical models (GGM) express conditional dependencies among variables of Gaussian-distributed high-dimensional data. However, real-life datasets exhibit heterogeneity which can be better captured through the use of mixtures of GGMs, where each component captures different conditional dependencies a.k.a. context-specific dependencies along with some common dependencies a.k.a. shared dependencies. Methods to discover shared and context-specific graphical structures include joint and grouped graphical Lasso, and the EM algorithm with various penalized likelihood scoring functions. However, these methods detect graphical structures with high false discovery rates and do not detect two types of dependencies (i.e., context-specific and shared) together. In this paper, we develop a method to discover shared conditional dependencies along with context-specific graphical models via a two-level hierarchical Gaussian graphical model. We assume that the graphical models corresponding to shared and context-specific dependencies are decomposable, which leads to an efficient greedy algorithm to select edges minimizing a score based on minimum message length (MML). The MML-based score results in lower false discovery rate, leading to a more effective structure discovery. We present extensive empirical results on synthetic and real-life datasets and show that our method leads to more accurate prediction of context-specific dependencies among random variables compared to previous works. Hence, we can consider that our method is a state of the art to discover both shared and context-specific conditional dependencies from high-dimensional Gaussian heterogeneous data.