Concave Penalized Estimation of Sparse Bayesian Networks

• Activity Number: 251
• Type: Contributed
• Date/Time: Monday, August 4, 2014 : 2:00 PM to 3:50 PM
• Sponsor: Section on Statistical Learning and Data Mining
• Author(s): Nikhyl Aragam*+ and Qing Zhou
• Companies: University of California, Los Angeles and University of California, Los Angeles
• Keywords: Bayesian networks ; Concave penalization ; Coordinate descent ; Nonconvex optimization ; Directed acyclic graphs ; Structural equation modeling

Abstract

We introduce a penalized likelihood approach to estimating the structure of a Gaussian Bayesian network, given by a directed acyclic graph, from observational data under a concave penalty. The framework introduced here does not rely on faithfulness or knowledge of the ordering of the variables and favours sparsity over complexity in estimating the underlying graph. Asymptotic theory for the estimator is provided in the finite-dimensional case, and a fast numerical scheme is offered that is capable of estimating the structure of graphs with thousands of nodes. Our algorithm also takes advantage of sparsity and acyclicity by using coordinate descent, a computational approach which has recently become quite popular. Finally, we have compared our method with the well-known PC algorithm, and can show that our method is faster in general and does a significantly better job of handling small samples and very sparse networks. Our focus is on the Gaussian linear model, however, the framework introduced here can also be extended to non-Gaussian and non-linear designs, which is an attractive prospect for future applications.