Open Access
June 2012 Half-trek criterion for generic identifiability of linear structural equation models
Rina Foygel, Jan Draisma, Mathias Drton
Ann. Statist. 40(3): 1682-1713 (June 2012). DOI: 10.1214/12-AOS1012


A linear structural equation model relates random variables of interest and corresponding Gaussian noise terms via a linear equation system. Each such model can be represented by a mixed graph in which directed edges encode the linear equations and bidirected edges indicate possible correlations among noise terms. We study parameter identifiability in these models, that is, we ask for conditions that ensure that the edge coefficients and correlations appearing in a linear structural equation model can be uniquely recovered from the covariance matrix of the associated distribution. We treat the case of generic identifiability, where unique recovery is possible for almost every choice of parameters. We give a new graphical condition that is sufficient for generic identifiability and can be verified in time that is polynomial in the size of the graph. It improves criteria from prior work and does not require the directed part of the graph to be acyclic. We also develop a related necessary condition and examine the “gap” between sufficient and necessary conditions through simulations on graphs with $25$ or $50$ nodes, as well as exhaustive algebraic computations for graphs with up to five nodes.


Download Citation

Rina Foygel. Jan Draisma. Mathias Drton. "Half-trek criterion for generic identifiability of linear structural equation models." Ann. Statist. 40 (3) 1682 - 1713, June 2012.


Published: June 2012
First available in Project Euclid: 2 October 2012

zbMATH: 1257.62059
MathSciNet: MR3015040
Digital Object Identifier: 10.1214/12-AOS1012

Primary: 62H05 , 62J05

Keywords: Covariance matrix , Gaussian distribution , Graphical model , multivariate normal distribution , parameter identification , structural equation model

Rights: Copyright © 2012 Institute of Mathematical Statistics

Vol.40 • No. 3 • June 2012
Back to Top