  Structural learning of directed acyclic graphs (DAGs) or Bayesian networks
has been studied extensively under the assumption that data are independent. We
propose a new Gaussian DAG model for dependent data which assumes the
observations are correlated according to an undirected network. Under this
model, we develop a method to estimate the DAG structure given a topological
ordering of the nodes. The proposed method jointly estimates the Bayesian
network and the correlations among observations by optimizing a scoring
function based on penalized likelihood. We show that under some mild
conditions, the proposed method produces consistent estimators after one
iteration. Extensive numerical experiments also demonstrate that by jointly
estimating the DAG structure and the sample correlation, our method achieves
much higher accuracy in structure learning. When the node ordering is unknown,
through experiments on synthetic and real data, we show that our algorithm can
be used to estimate the correlations between samples, with which we can
de-correlate the dependent data to significantly improve the performance of
classical DAG learning methods.
