Cross-covariance modelling via DAGs with hidden variables

TL;DR

This paper characterizes the set of distributions generated by Gaussian latent variable models relating two observed variable blocks, focusing on cross-covariance, and explores their identifiability and covariance equivalence.

Contribution

It provides an exact characterization of distributions in Gaussian latent variable models for cross-covariance, including novel covariance equivalence results and alternative parametrizations.

Findings

Model relates to singular value decomposition of cross-covariance

Despite underidentification, useful information can be extracted

Provides covariance equivalence results for Gaussian hidden variable models

Abstract

DAG models with hidden variables present many difficulties that are not present when all nodes are observed. In particular, fully observed DAG models are identified and correspond to well-defined sets ofdistributions, whereas this is not true if nodes are unobserved. Inthis paper we characterize exactly the set of distributions given by a class of one-dimensional Gaussian latent variable models. 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 decomposition of the cross-covariance matrix. We show that, although the model is underidentified, useful information may be extracted. We further consider an alternative parametrization in which one latent variable is associated with each block. Our analysis leads to some novel covariance equivalence results for Gaussian hidden variable models.