Causal generalized linear models via Pearson risk invariance

Prediction invariance of causal models under heterogeneous settings has been exploited by a number of recent methods for causal discovery, typically focussing on recovering the causal parents of a target variable of interest. Existing methods require observational data from a number of sufficiently different environments, which is rarely available. In this paper, we consider a structural equation model where the target variable is described by a generalized linear model conditional on its parents. Besides having finite moments, no modelling assumptions are made on the conditional distributions of the other variables in the system, and nonlinear effects on the target variable can naturally be accommodated by a generalized additive structure. Under this setting, we characterize the causal model uniquely by means of two key properties: the Pearson risk invariant under the causal model and, conditional on the causal parents, the causal parameters maximize the expected likelihood. These two properties form the basis of a computational strategy for searching the causal model among all possible models. A stepwise greedy search is proposed for systems with a large number of variables. Crucially, for generalized linear models with a known dispersion parameter, such as Poisson and logistic regression, the causal model can be identified from a single data environment. The method is implemented in the R package causalreg.