  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 article addresses the difficult task of quantifying graph
uncertainty by developing a variational Bayes 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 constraint. We empirically demonstrate that our method, ProDAG, can
deliver accurate inference, and often outperforms existing state-of-the-art
alternatives.
