Cross-covariance modelling via DAGs with hidden variables

DAG models with hidden variables present many difficulties that are absent when all nodes are observed. In particular, fully ob­ served DAG models are identified and cor­ respond to well-defined sets of distributions, whereas this is not true if nodes are unob­ served. In this paper we characterize exactly the set of distributions given by a class of Gaussian models with one-dimensional latent variables. These models relate two blocks of observed variables, modeling only the cross­ covariance matrix. We describe the relation of this model to the singular value decom­ position of the cross-covariance matrix. We show that, although the model is underiden­ tified, useful information may be extracted. We further consider an alternative parame­ terization in which one latent variable is as­ sociated with each block. Our analysis leads to some novel covariance equivalence results for Gaussian hidden variable models.