  An important problem in many domains is to predict how a system will respond
to interventions. This task is inherently linked to estimating the system's
underlying causal structure. To this end, Invariant Causal Prediction (ICP)
(Peters et al., 2016) has been proposed which learns a causal model exploiting
the invariance of causal relations using data from different environments. When
considering linear models, the implementation of ICP is relatively
straightforward. However, the nonlinear case is more challenging due to the
difficulty of performing nonparametric tests for conditional independence. In
this work, we present and evaluate an array of methods for nonlinear and
nonparametric versions of ICP for learning the causal parents of given target
variables. We find that an approach which first fits a nonlinear model with
data pooled over all environments and then tests for differences between the
residual distributions across environments is quite robust across a large
variety of simulation settings. We call this procedure "invariant residual
distribution test". In general, we observe that the performance of all
approaches is critically dependent on the true (unknown) causal structure and
it becomes challenging to achieve high power if the parental set includes more
than two variables. As a real-world example, we consider fertility rate
modelling which is central to world population projections. We explore
predicting the effect of hypothetical interventions using the accepted models
from nonlinear ICP. The results reaffirm the previously observed central causal
role of child mortality rates.
