  Directed Gaussian graphical models are statistical models that use a directed
acyclic graph (DAG) to represent the conditional independence structures
between a set of jointly normal random variables. The DAG specifies the model
through recursive factorization of the parametrization, via restricted
conditional distributions. In this paper, we make an attempt to characterize
the DAGs whose vanishing ideals are toric ideals. In particular, we give some
combinatorial criteria to construct such DAGs from smaller DAGs which have
toric vanishing ideals. An associated monomial map called the shortest trek map
plays an important role in our description of toric Gaussian DAG models. For
DAGs whose vanishing ideal is toric, we prove results about the generating sets
of those toric ideals.
