Learning Sparse Gaussian Bayesian Network Structure by Variable Grouping

Bayesian networks (BNs) are popular for modeling conditional distributions of variables and causal relationships, especially in biological settings such as protein interactions, gene regulatory networks and microbial interactions. Previous BN structure learning algorithms treat variables with similar tendency separately. In this paper, we propose a grouped sparse Gaussian BN (GSGBN) structure learning algorithm which creates BN based on three assumptions: (i) variables follow a multivariate Gaussian distribution, (ii) the network only contains a few edges (sparse), (iii) similar variables have less-divergent sets of parents, while not-so-similar ones should have divergent sets of parents (variable grouping). We use L₁ regularization to make the learned network sparse, and another term to incorporate shared information among variables. For similar variables, GSGBN tends to penalize the differences of similar variables´ parent sets more, compared to those not-so-similar variables´ parent sets. The similarity of variables is learned from the data by alternating optimization, without prior domain knowledge. Based on this new definition of the optimal BN, a coordinate descent algorithm and a projected gradient descent algorithm are developed to obtain edges of the network and also similarity of variables. Experimental results on both simulated and real datasets show that GSGBN has substantially superior prediction performance for structure learning when compared to several existing algorithms.