Half-Trek Criterion for Identifiability of Latent Variable Models

TL;DR

This paper introduces the latent-factor half-trek criterion (LF-HTC), a new method for certifying the identifiability of direct effects in linear structural equation models with latent variables, especially effective in dense effect scenarios.

Contribution

The paper develops the LF-HTC, a novel criterion operating on the original model to certify identifiability even with dense latent effects, improving over prior methods.

Findings

LF-HTC certifies identifiability in complex models.

The criterion is polynomial-time verifiable with bounded latent subset size.

It enables recovery of effects as rational functions of observed covariances.

Abstract

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.