Open Access
December 2022 Half-trek criterion for identifiability of latent variable models
Rina Foygel Barber, Mathias Drton, Nils Sturma, Luca Weihs
Author Affiliations +
Ann. Statist. 50(6): 3174-3196 (December 2022). DOI: 10.1214/22-AOS2221


We consider linear structural equation models with latent variables and develop a criterion to certify whether the direct causal effects between the observable variables are identifiable based on the observed covariance matrix. Linear structural equation models assume that both observed and latent variables solve a linear equation system featuring stochastic noise terms. Each model corresponds to a directed graph whose edges represent the direct effects that appear as coefficients in the equation system. Prior research has developed a variety of methods to decide identifiability of direct effects in a latent projection framework, in which the confounding effects of the latent variables are represented by correlation among noise terms. This approach is effective when the confounding is sparse and effects only small subsets of the observed variables. In contrast, the new latent-factor half-trek criterion (LF-HTC) we develop in this paper operates on the original unprojected latent variable model and is able to certify identifiability in settings, where some latent variables may also have dense effects on many or even all of the observables. Our LF-HTC is an effective sufficient criterion for rational identifiability, under which the direct effects can be uniquely recovered as rational functions of the joint covariance matrix of the observed random variables. When restricting the search steps in LF-HTC to consider subsets of latent variables of bounded size, the criterion can be verified in time that is polynomial in the size of the graph.

Funding Statement

This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 883818).
Nils Sturma acknowledges support by the MDSI via the Linde/MDSI PhD Fellowship program.
Rina Foygel Barber was supported by the U.S. National Science Foundation via grants DMS-1654076 and DMS-2023109, and by the Office of Naval Research via grant N00014-20-1-2337.


Mathias Drton and Nils Sturma are affiliated with the Munich Data Science Institute (MDSI) and the Munich Center for Machine Learning (MCML).


Download Citation

Rina Foygel Barber. Mathias Drton. Nils Sturma. Luca Weihs. "Half-trek criterion for identifiability of latent variable models." Ann. Statist. 50 (6) 3174 - 3196, December 2022.


Received: 1 January 2022; Revised: 1 June 2022; Published: December 2022
First available in Project Euclid: 21 December 2022

MathSciNet: MR4524493
zbMATH: 07641122
Digital Object Identifier: 10.1214/22-AOS2221

Primary: 62H22 , 62J05 , 62R01

Keywords: Covariance matrix , factor analysis , Graphical model , hidden variables , latent variables , parameter identification , structural equation model

Rights: Copyright © 2022 Institute of Mathematical Statistics

Vol.50 • No. 6 • December 2022
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