  In multivariate statistics, acyclic mixed graphs with directed and bidirected
edges are widely used for compact representation of dependence structures that
can arise in the presence of hidden (i.e., latent or unobserved) variables.
Indeed, under multivariate normality, every mixed graph corresponds to a set of
covariance matrices that contains as a full-dimensional subset the covariance
matrices associated with a causally interpretable acyclic digraph. This digraph
generally has some of its nodes corresponding to hidden variables. We seek to
clarify for which mixed graphs there exists an acyclic digraph whose hidden
variable model coincides with the mixed graph model. Restricting to the
tractable setting of chain graphs and multivariate normality, we show that
decomposability of the bidirected part of the chain graph is necessary and
sufficient for equality between the mixed graph model and some hidden variable
model given by an acyclic digraph.
