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.
翻译:在数据是独立的假设下,对定向单流图(DAGs)或巴伊西亚网络的结构学习进行了广泛研究。我们提议了一个新的Gaussian DAG模型,用于根据非定向网络进行相关观测的附属数据。在这个模型下,我们开发了一种方法,根据节点的地形顺序来估计DAG结构的结构。拟议方法通过根据受罚可能性优化评分功能,共同估计贝伊西亚网络和观测结果之间的相互关系。我们显示,在一些温和条件下,拟议方法产生经过一次迭代后一致的估测数据。广泛的数字实验还表明,通过共同估计DAG结构和抽样相关性,我们的方法在结构学习中实现了更高的准确性。当节点的排序未知时,我们通过合成和真实数据的实验,我们表明,我们的算法可以用来估计样本之间的关联性,我们可以据此对依赖数据进行分解,从而大大改进经典DAG学习方法的性能。