On statistical and causal models associated with acyclic directed mixed graphs

TL;DR

This paper advocates for the noise expansion (NE) interpretation of acyclic directed mixed graphs (ADMGs) in causal modeling, demonstrating its theoretical advantages and unifying properties over other interpretations, and developing an ADMG-based causality framework.

Contribution

It establishes the NE model as the default interpretation for ADMGs, proves its completeness and nested Markov property, and compares it favorably against the latent DAG interpretation.

Findings

The NE model is equivalent to other interpretations in unconfounded graphs.

The NE model for any ADMG is the union of NE models for its unconfounded expansions.

The NE model is shown to be nested Markov, supporting causal inference.

Abstract

Causal models in statistics are often described using acyclic directed mixed graphs (ADMGs), which contain directed and bidirected edges and no directed cycles. This article surveys various interpretations of ADMGs, discusses their relations in different sub-classes of ADMGs, and argues that one of them -- the noise expansion (NE) model -- should be used as the default interpretation. Our endorsement of the NE model is based on two observations. First, in a subclass of ADMGs called unconfounded graphs (which retain most of the good properties of directed acyclic graphs and bidirected graphs), the NE model is equivalent to many other interpretations including the global Markov and nested Markov models. Second, the NE model for an arbitrary ADMG is exactly the union of that for all unconfounded expansions of that graph. This property is referred to as completeness, as it shows that the model does not commit to any specific latent variable explanation. In proving that the NE model is nested Markov, we also develop an ADMG-based theory for causality. Finally, we compare the NE model with the closely related but different interpretation of ADMGs as directed acyclic graphs (DAGs) with latent variables that is commonly used in the literature. We argue that the "latent DAG" interpretation is mathematically unnecessary, makes obscure ontological assumptions, and discourages practitioners from deliberating over important structural assumptions.