# Nested Markov Properties for Acyclic Directed Mixed Graphs

## TL;DR

This paper introduces a new framework for understanding conditional independence and equality constraints in acyclic directed mixed graphs (ADMGs), extending DAG models and aiding causal effect identification.

## Contribution

It develops a novel approach to characterize ADMG models using fixing operations, unifying conditional independences and equality constraints like the Verma constraint.

## Key findings

- ADMG models can be characterized by new Markov properties and factorizations.
- Verma constraints are interpretable as conditional independences in kernel objects.
- The fixing operation simplifies the identification of causal effects in hidden variable models.

## Abstract

Conditional independence models associated with directed acyclic graphs (DAGs) may be characterized in at least three different ways: via a factorization, the global Markov property (given by the d-separation criterion), and the local Markov property. Marginals of DAG models also imply equality constraints that are not conditional independences; the well-known ``Verma constraint'' is an example. Constraints of this type are used for testing edges, and in a computationally efficient marginalization scheme via variable elimination. We show that equality constraints like the ``Verma constraint'' can be viewed as conditional independences in kernel objects obtained from joint distributions via a fixing operation that generalizes conditioning and marginalization. We use these constraints to define, via ordered local and global Markov properties, and a factorization, a graphical model associated with acyclic directed mixed graphs (ADMGs). We prove that marginal distributions of DAG models lie in this model, and that a set of these constraints given by Tian provides an alternative definition of the model. Finally, we show that the fixing operation used to define the model leads to a particularly simple characterization of identifiable causal effects in hidden variable causal DAG models.