  In this paper, we study the high-dimensional sparse directed acyclic graph
(DAG) models under the empirical sparse Cholesky prior. Among our results,
strong model selection consistency or graph selection consistency is obtained
under more general conditions than those in the existing literature. Compared
to Cao, Khare and Ghosh (2017), the required conditions are weakened in terms
of the dimensionality, sparsity and lower bound of the nonzero elements in the
Cholesky factor. Furthermore, our result does not require the irrepresentable
condition, which is necessary for Lasso type methods. We also derive the
posterior convergence rates for precision matrices and Cholesky factors with
respect to various matrix norms. The obtained posterior convergence rates are
the fastest among those of the existing Bayesian approaches. In particular, we
prove that our posterior convergence rates for Cholesky factors are the minimax
or at least nearly minimax depending on the relative size of true sparseness
for the entire dimension. The simulation study confirms that the proposed
method outperforms the competing methods.
