  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.
