ProDAG: Projection-Induced Variational Inference for Directed Acyclic Graphs

Directed acyclic graph (DAG) learning is a rapidly expanding field of research. Though the field has witnessed remarkable advances over the past few years, it remains statistically and computationally challenging to learn a single (point estimate) DAG from data, let alone provide uncertainty quantification. Our paper addresses the difficult task of quantifying graph uncertainty by developing a Bayesian variational inference framework based on novel distributions that have support directly on the space of DAGs. The distributions, which we use to form our prior and variational posterior, are induced by a projection operation, whereby an arbitrary continuous distribution is projected onto the space of sparse weighted acyclic adjacency matrices (matrix representations of DAGs) with probability mass on exact zeros. Though the projection constitutes a combinatorial optimization problem, it is solvable at scale via recently developed techniques that reformulate acyclicity as a continuous onstraint. We empirically demonstrate that our proposed method, ProDAG, can perform higher quality Bayesian inference than possible with existing state-of-the-art alternatives.