Identifiability of directed Gaussian graphical models with one latent source

TL;DR

This paper investigates the conditions under which parameters in directed Gaussian graphical models with a single latent confounder are identifiable, providing graphical criteria and computational validation for models with up to six observed variables.

Contribution

It introduces new graphical conditions for both sufficient and necessary local identifiability of such models, and relates the identifiability of larger models to their subgraphs.

Findings

Graphical conditions for sufficient and necessary identifiability.

Identifiability can be inferred from subgraph models.

Computational study confirms criteria for models with 4-6 variables.

Abstract

We study parameter identifiability of directed Gaussian graphical models with one latent variable. In the scenario we consider, the latent variable is a confounder that forms a source node of the graph and is a parent to all other nodes, which correspond to the observed variables. We give a graphical condition that is sufficient for the Jacobian matrix of the parametrization map to be full rank, which entails that the parametrization is generically finite-to-one, a fact that is sometimes also referred to as local identifiability. We also derive a graphical condition that is necessary for such identifiability. Finally, we give a condition under which generic parameter identifiability can be determined from identifiability of a model associated with a subgraph. The power of these criteria is assessed via an exhaustive algebraic computational study on models with 4, 5, and 6 observable variables.