Identifiability of Generalized Hypergeometric Distribution (GHD) Directed Acyclic Graphical Models

We introduce a new class of identifiable DAG models where the conditional distribution of each node given its parents belongs to a family of generalized hypergeometric distributions (GHD). A family of generalized hypergeometric distributions includes a lot of discrete distributions such as the binomial, Beta-binomial, negative binomial, Poisson, hyper-Poisson, and many more. We prove that if the data drawn from the new class of DAG models, one can fully identify the graph structure. We further present a reliable and polynomial-time algorithm that recovers the graph from finitely many data. We show through theoretical results and numerical experiments that our algorithm is statistically consistent in high-dimensional settings (p > n) if the indegree of the graph is bounded, and out-performs state-of-the-art DAG learning algorithms.