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

On the causal interpretation of acyclic mixed graphs under multivariate normality