February 2023 Nested Markov properties for acyclic directed mixed graphs
Thomas S. Richardson, Robin J. Evans, James M. Robins, Ilya Shpitser
Author Affiliations +
Ann. Statist. 51(1): 334-361 (February 2023). DOI: 10.1214/22-AOS2253


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.

Funding Statement

The first author was supported in part by ONR Grants N00014-19-1-2446 and N00014-15-1-2672, and NIH Grant R01 AI032475.
The third author was supported in part by ONR Grant N00014-19-1-2446, and NIH Grant R01 AI032475.
The fourth author was supported in part by ONR Grant N00014-21-1-2820, NSF Grants 2040804 and 1942239, and NIH Grant R01 AI127271-01A1.


We thank Zhongyi Hu for pointing out an error in the proof of Proposition 6, and the Associate Editor and referees for their helpful comments. The authors completed work on this paper while visiting the American Institute for Mathematics and the Simons Institute, Berkeley, California.


Download Citation

Thomas S. Richardson. Robin J. Evans. James M. Robins. Ilya Shpitser. "Nested Markov properties for acyclic directed mixed graphs." Ann. Statist. 51 (1) 334 - 361, February 2023. https://doi.org/10.1214/22-AOS2253


Received: 1 April 2022; Revised: 1 November 2022; Published: February 2023
First available in Project Euclid: 23 March 2023

MathSciNet: MR4564859
zbMATH: 07684015
Digital Object Identifier: 10.1214/22-AOS2253

Primary: 62H99
Secondary: 60E05

Keywords: Causal inference , Conditional independence , graphical models , hidden variable models

Rights: Copyright © 2023 Institute of Mathematical Statistics


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Vol.51 • No. 1 • February 2023
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