Identification of causal direction between a causal-effect pair from observed
data has recently attracted much attention. Various methods based on functional
causal models have been proposed to solve this problem, by assuming the causal
process satisfies some (structural) constraints and showing that the reverse
direction violates such constraints. The nonlinear additive noise model has
been demonstrated to be effective for this purpose, but the model class is not
transitive--even if each direct causal relation follows this model, indirect
causal influences, which result from omitted intermediate causal variables and
are frequently encountered in practice, do not necessarily follow the model
constraints; as a consequence, the nonlinear additive noise model may fail to
correctly discover causal direction. In this work, we propose a cascade
nonlinear additive noise model to represent such causal influences--each direct
causal relation follows the nonlinear additive noise model but we observe only
the initial cause and final effect. We further propose a method to estimate the
model, including the unmeasured intermediate variables, from data, under the
variational auto-encoder framework. Our theoretical results show that with our
model, causal direction is identifiable under suitable technical conditions on
the data generation process. Simulation results illustrate the power of the
proposed method in identifying indirect causal relations across various
settings, and experimental results on real data suggest that the proposed model
and method greatly extend the applicability of causal discovery based on
functional causal models in nonlinear cases.

提出了一种级联非线性加性噪声模型，可以发现难以被非线性加性噪声模型正确揭示的因果关系。该研究还在变分自编码器框架下提出一种方法来估计该模型并克服了实践中存在的一些问题。