  We study a notion of positivity of Gaussian directed acyclic graphical models
corresponding to a non-negativity constraint on the coefficients of the
associated structural equation model. We prove that this constraint is
equivalent to the distribution being conditionally increasing in sequence
(CIS), a well-known subclass of positively associated random variables. These
distributions require knowledge of a permutation, a CIS ordering, of the nodes
for which the constraint of non-negativity holds. We provide an algorithm and
prove in the noise-less setting that a CIS ordering can be recovered when it
exists. We extend this result to the noisy setting and provide assumptions for
recovering the CIS orderings. In addition, we provide a characterization of
Markov equivalence for CIS DAG models. Further, we show that when a CIS
ordering is known, the corresponding class of Gaussians lies in a family of
distributions in which maximum likelihood estimation is a convex problem.
