High-Dimensional Poisson DAG Model Learning Using ℓ1-Regularized Regression

In this paper, we develop a new approach to learning high-dimensional Poisson directed acyclic graphical (DAG) models from only observational data without strong assumptions such as faithfulness and strong sparsity. A key component of our method is to decouple the ordering estimation or parent search where the problems can be efficiently addressed using ℓ1-regularized regression and the mean-variance relationship. We show that sample size n = Omega( d2 log9 p) is sufficient for our polynomial time Mean-variance Ratio Scoring (MRS) algorithm to recover the true directed graph, where p is the number of nodes and d is the maximum indegree. We verify through simulations that our algorithm is statistically consistent in the high-dimensional p>n setting, and performs well compared to state-of-the-art ODS, GES, and MMHC algorithms. We also demonstrate through multivariate real count data that our MRS algorithm is well-suited to estimating DAG models for multivariate count data in comparison to other methods used for discrete data.