  We study parameter identifiability of directed Gaussian graphical models with
one latent variable. In the scenario we consider, the latent variable is a
confounder that forms a source node of the graph and is a parent to all other
nodes, which correspond to the observed variables. We give a graphical
condition that is sufficient for the Jacobian matrix of the parametrization map
to be full rank, which entails that the parametrization is generically
finite-to-one, a fact that is sometimes also referred to as local
identifiability. We also derive a graphical condition that is necessary for
such identifiability. Finally, we give a condition under which generic
parameter identifiability can be determined from identifiability of a model
associated with a subgraph. The power of these criteria is assessed via an
exhaustive algebraic computational study on models with 4, 5, and 6 observable
variables.
