Cross-Covariance Modelling via DAGs with Hidden Variables

Abstract

DAG models with hidden variables present many difficulties that are absent when all nodes are observed. In particular, fully observed DAG models are identified and correspond to well-defined sets of distributions, whereas this is not true if nodes are unobserved. 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 crosscovariance matrix. We describe the relation of this model to the singular value decomposition of the cross-covariance matrix. We show that, although the model is underidentified, useful information may be extracted. We further consider an alternative parameterization in which one latent variable is associated with each block. Our analysis leads to some novel covariance equivalence results for Gaussian hidden variable models.