  We propose an empirical Bayes formulation of the structure learning problem,
where the prior specification assumes that all node variables have the same
error variance, an assumption known to ensure the identifiability of the
underlying causal directed acyclic graph (DAG). To facilitate efficient
posterior computation, we approximate the posterior probability of each
ordering by that of a best DAG model, which naturally leads to an order-based
Markov chain Monte Carlo (MCMC) algorithm. Strong selection consistency for our
model in high-dimensional settings is proved under a condition that allows
heterogeneous error variances, and the mixing behavior of our sampler is
theoretically investigated. Further, we propose a new iterative top-down
algorithm, which quickly yields an approximate solution to the structure
learning problem and can be used to initialize the MCMC sampler. We demonstrate
that our method outperforms other state-of-the-art algorithms under various
simulation settings, and conclude the paper with a single-cell real-data study
illustrating practical advantages of the proposed method.
