ABSTRACT
This chapter describes a simple formulation of identification theory for common targets of inference that arise in causal inference, developed in the context of non-parametric graphical causal models. It discusses extensions of this theory to an important type of causal model where counterfactual random variables are determined via linear causal mechanisms and Gaussian noise. These models are known as linear structural equation models with correlated errors. The chapter provides a characterization of identifiable targets of causal inference in hidden variable causal directed acyclic graphs (DAG), and identification algorithms that yield appropriate generalizations of the g-formula. The latent projection acyclic directed mixed graph represents an infinite class of hidden variable DAGs that all share identification theory. In causal models represented by DAGs where all relevant variables are observed, counterfactual responses to interventions that set variables to constants or via a known function of other variables are identified by versions of the g-formula.
