Abstract
Identification of causal direction between a causaleffect 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 autoencoder 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