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.

我们研究了在存在潜在变量的条件下学习因果结构的复杂任务，包括定位潜在变量和确定它们的数量，以及确定潜在和观察变量之间的因果关系。我们提出了一种用于线性非高斯非循环因果模型（包括潜在变量）的广义独立噪声（GIN）条件，该条件确定了某些测量变量的线性组合与其他测量变量之间的独立性。通过与 GIN 的图形准则的比较，我们发现独立噪声条件可以看作是 GIN 的特殊情况。借助所提出的 GIN 条件和精心设计的搜索程序，我们进一步高效地估计了线性非高斯潜在层次模型（LiNGLaHs），其中潜在变量可能具有因果关系，甚至可能遵循层次结构。实验结果证明了所提出方法的有效性。