  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 $\ell_1$-regularized regression and
the mean-variance relationship. We show that sample size $n = \Omega( d^{2}
\log^{9} p)$ 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.
