Hierarchical Modeling of Directed Acyclic Graphs: Estimation, Selection and Asymptotics

This proposed research project will develop efficient algorithms for estimating large-scale directed acyclic graphs (DAGs) based on a novel concept of topological layer. DAG is a popularly used structure in causal inference to encode joint distribution of a randomvector through nodes and directed edge. One key challenge of DAG estimation is due to the unknown order of nodes, which amounts to a discrete space of DAG structures whose cardinality is of exponential order of the number of nodes. The proposed conceptof topological layer introduces a natural way to determine the relative order of nodes in DAG and greatly facilitates the estimation of large-scale DAGs. On this ground, the PI will develop efficient estimation algorithms based on various model assumptions,including linear and nonlinear structural equation model (SEM). The developed methods mainly consist of determination of topological layers and then estimation of directed edges for parent-child relations. The PI will investigate the theoretical properties of theproposed methods, and establish their asymptotic and finite-sample probability bounds. The PI will also develop efficient computing algorithms to facilitate large-scale estimation, integrating the strength of parallel computing platform. The proposed methods will beapplied to understand the causal relationship in gene interaction networks.