Generalized Independent Noise Condition for Estimating Causal Structure
  with Latent Variables
Feng Xie Biwei Huang Zhengming Chen Ruichu Cai Clark Glymour Zhi Geng Kun Zhang
We investigate the challenging task of learning causal structure in the
presence of latent variables, including locating latent variables and
determining their quantity, and identifying causal relationships among both
latent and observed variables. To address this, we propose a Generalized
Independent Noise (GIN) condition for linear non-Gaussian acyclic causal models
that incorporate latent variables, which establishes the independence between a
linear combination of certain measured variables and some other measured
variables. Specifically, for two observed random vectors $\bf{Y}$ and $\bf{Z}$,
GIN holds if and only if $\omega^{\intercal}\mathbf{Y}$ and $\mathbf{Z}$ are
independent, where $\omega$ is a non-zero parameter vector determined by the
cross-covariance between $\mathbf{Y}$ and $\mathbf{Z}$. We then give necessary
and sufficient graphical criteria of the GIN condition in linear non-Gaussian
acyclic causal models. Roughly speaking, GIN implies the existence of an
exogenous set $\mathcal{S}$ relative to the parent set of $\mathbf{Y}$ (w.r.t.
the causal ordering), such that $\mathcal{S}$ d-separates $\mathbf{Y}$ from
$\mathbf{Z}$. Interestingly, we find that the independent noise condition
(i.e., if there is no confounder, causes are independent of the residual
derived from regressing the effect on the causes) can be seen as a special case
of GIN. With such a connection between GIN and latent causal structures, we
further leverage the proposed GIN condition, together with a well-designed
search procedure, to efficiently estimate Linear, Non-Gaussian Latent
Hierarchical Models (LiNGLaHs), where latent confounders may also be causally
related and may even follow a hierarchical structure. We show that the
underlying causal structure of a LiNGLaH is identifiable in light of GIN
conditions under mild assumptions. Experimental results show the effectiveness
of the proposed approach.