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Learning DAG or Bayesian network models is an important problem inmulti-variate causal inference. However, a number of challenges arises inlearning large-scale DAG models including model identifiability andcomputational complexity since the space of directed graphs is huge. In thispaper, we address these issues in a number of steps for a broad class of DAGmodels where the noise or variance is signal-dependent. Firstly we introduce anew class of identifiable DAG models, where each node has a distribution wherethe variance is a quadratic function of the mean (QVF DAG models). Our QVF DAGmodels include many interesting classes of distributions such as Poisson,Binomial, Geometric, Exponential, Gamma and many other distributions in whichthe noise variance depends on the mean. We prove that this class of QVF DAGmodels is identifiable, and introduce a new algorithm, the OverDispersionScoring (ODS) algorithm, for learning large-scale QVF DAG models. Our algorithmis based on firstly learning the moralized or undirected graphical modelrepresentation of the DAG to reduce the DAG search-space, and then exploitingthe quadratic variance property to learn the causal ordering. We show throughtheoretical results and simulations that our algorithm is statisticallyconsistent in the high-dimensional p>n setting provided that the degree of themoralized graph is bounded and performs well compared to state-of-the-artDAG-learning algorithms.

Learning Quadratic Variance Function (QVF) DAG models via OverDispersion Scoring (ODS)