# Invariant Causal Prediction for Nonlinear Models

## TL;DR

This paper develops and evaluates nonlinear, nonparametric methods for Invariant Causal Prediction (ICP), demonstrating a robust residual distribution test approach for identifying causal parents across diverse settings and applying it to fertility rate modeling.

## Contribution

It introduces a new nonlinear, nonparametric ICP method called invariant residual distribution test and evaluates its robustness and effectiveness in various simulation and real-world scenarios.

## Key findings

- The residual distribution test approach is robust across many simulations.
- Performance depends on the true causal structure, especially with larger parental sets.
- Application to fertility modeling confirms causal influence of child mortality rates.

## Abstract

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.