## Bernoulli

- Bernoulli
- Volume 19, Number 5B (2013), 2330-2358.

### Stable mixed graphs

#### Abstract

In this paper, we study classes of graphs with three types of edges that capture the modified independence structure of a directed acyclic graph (DAG) after marginalisation over unobserved variables and conditioning on selection variables using the $m$-separation criterion. These include MC, summary, and ancestral graphs. As a modification of MC graphs, we define the class of ribbonless graphs (RGs) that permits the use of the $m$-separation criterion. RGs contain summary and ancestral graphs as subclasses, and each RG can be generated by a DAG after marginalisation and conditioning. We derive simple algorithms to generate RGs, from given DAGs or RGs, and also to generate summary and ancestral graphs in a simple way by further extension of the RG-generating algorithm. This enables us to develop a parallel theory on these three classes and to study the relationships between them as well as the use of each class.

#### Article information

**Source**

Bernoulli, Volume 19, Number 5B (2013), 2330-2358.

**Dates**

First available in Project Euclid: 3 December 2013

**Permanent link to this document**

https://projecteuclid.org/euclid.bj/1386078605

**Digital Object Identifier**

doi:10.3150/12-BEJ454

**Mathematical Reviews number (MathSciNet)**

MR3160556

**Zentralblatt MATH identifier**

1317.05077

**Keywords**

ancestral graph directed acyclic graph independence model $m$-separation criterion marginalisation and conditioning MC graph summary graph

#### Citation

Sadeghi, Kayvan. Stable mixed graphs. Bernoulli 19 (2013), no. 5B, 2330--2358. doi:10.3150/12-BEJ454. https://projecteuclid.org/euclid.bj/1386078605