  Directed graphical models specify noisy functional relationships among a
collection of random variables. In the Gaussian case, each such model
corresponds to a semi-algebraic set of positive definite covariance matrices.
The set is given via parametrization, and much work has gone into obtaining an
implicit description in terms of polynomial (in-)equalities. Implicit
descriptions shed light on problems such as parameter identification, model
equivalence, and constraint-based statistical inference. For models given by
directed acyclic graphs, which represent settings where all relevant variables
are observed, there is a complete theory: All conditional independence
relations can be found via graphical $d$-separation and are sufficient for an
implicit description. The situation is far more complicated, however, when some
of the variables are hidden (or in other words, unobserved or latent). We
consider models associated to mixed graphs that capture the effects of hidden
variables through correlated error terms. The notion of trek separation
explains when the covariance matrix in such a model has submatrices of low rank
and generalizes $d$-separation. However, in many cases, such as the infamous
Verma graph, the polynomials defining the graphical model are not
determinantal, and hence cannot be explained by $d$-separation or
trek-separation. In this paper, we show that these constraints often correspond
to the vanishing of nested determinants and can be graphically explained by a
notion of restricted trek separation.
