## The Annals of Statistics

- Ann. Statist.
- Volume 40, Number 3 (2012), 1682-1713.

### Half-trek criterion for generic identifiability of linear structural equation models

Rina Foygel, Jan Draisma, and Mathias Drton

#### Abstract

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.

#### Article information

**Source**

Ann. Statist., Volume 40, Number 3 (2012), 1682-1713.

**Dates**

First available in Project Euclid: 2 October 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1349196388

**Digital Object Identifier**

doi:10.1214/12-AOS1012

**Mathematical Reviews number (MathSciNet)**

MR3015040

**Zentralblatt MATH identifier**

1257.62059

**Subjects**

Primary: 62H05: Characterization and structure theory 62J05: Linear regression

**Keywords**

Covariance matrix Gaussian distribution graphical model multivariate normal distribution parameter identification structural equation model

#### Citation

Foygel, Rina; Draisma, Jan; Drton, Mathias. Half-trek criterion for generic identifiability of linear structural equation models. Ann. Statist. 40 (2012), no. 3, 1682--1713. doi:10.1214/12-AOS1012. https://projecteuclid.org/euclid.aos/1349196388

#### Supplemental materials

- Supplementary material: Inconclusive graphs, proofs and algorithms. The supplement starts with lists of some mixed graphs on $m=5$ nodes that are not classifiable using our methods, to illustrate the existing “gap” between our two criteria. After that we prove lemmas used in the main paper for establishing the HTC-identifiability and HTC-infinite-to-one criteria, and we provide details for the results relating HTC-identifiability to GC-identifiability and to graph decomposition. We then give correctness proofs for our algorithms for checking the HTC-criteria, and we discuss the weak HTC-criteria. The supplementary article concludes with a computational-algebraic discussion of the polynomial equations that led to the HTC-criteria.Digital Object Identifier: doi:10.1214/12-AOS1012SUPP