Discrete Dynamics in Nature and Society

Volume 2015, Article ID 982495, 5 pages

http://dx.doi.org/10.1155/2015/982495

## Local Structure Recovery of Chain Graphs after Marginalization

^{1}School of Mathematical Sciences, Shandong Normal University, Jinan 250014, China^{2}School of Mathematical Sciences, Peking University, Bejing 100871, China

Received 15 October 2014; Accepted 6 March 2015

Academic Editor: Bo Yang

Copyright © 2015 Qiang Zhao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper discusses local structure recovery of chain graphs (CGs) when there exist unobserved or latent variables or after marginalization over observed variables. Under a condition presented in this paper, it is explained which edges and directions of edges in local structure can be recovered validly and which cannot after marginalization.

#### 1. Introduction

Graphical models, also known as Markov networks and Bayesian networks, including independence graphs, directed acyclic graphs (DAGs), and chain graphs (CGs) have been applied widely to many fields, such as stochastic systems, data mining, pattern recognition, artificial intelligence, and causal discovery. Chain graphs (CGs) are widely used to represent independence, conditional independence, and causal relationships among the random variables [1–5]. Structure recovery of CGs has been discussed by many authors [4–6]. A statistical conclusion may be reversed after marginalization over some variables, which is called Yule-Simpson paradox [7, 8]. For sampling design, prior knowledge or assumptions on models are necessary for valid structural learning of CGs and parameter estimates since some variables may be unobserved, such as the faithfulness assumption [4, 5] and collapsibility [9, 10]. On the other hand, for data analysis, conditions are necessary for marginalizing over some observed variables. Various conditions have been presented for avoiding a statistical conclusion reversion about association and parameters of linear models [10–14]. Collapsibility of parameter estimates for undirected graphical models over some variables has been discussed in [9, 15–17], and that of DAGs has been discussed in [18, 19].

In this paper, we discuss local structure recovery for a CG when there exist unobserved or latent variables or after marginalization over observed variables. Suppose that there is an unknown true CG with a large number of variables but we may be interested in construction of a local structure of the CG from a marginal distribution of a subset of variables. We present a condition for this localized recovery and explain which edges and directions of edges in local structure can be recovered validly from the marginal distribution and which edges may be spurious. We say that an edge or a direction is recovered validly from the marginal distribution if it is the same as that recovered from the joint distribution. The condition is useful for both sampling design and data analysis. This localization of structure recovery is related to identification and collapsibility.

Section 2 gives notation and definitions. In Section 3, we present theoretical results on local structure recovery. Finally a discussion is given in Section 4.

#### 2. Notation and Definitions

In this section, we briefly introduce terminologies and notations on graph theory. Readers can refer to [3, 20] for more details.

Let be a graph with a vertex set and an edge set . We say that there is an undirected edge, or a line, between vertex and vertex (denoted as ), if and ; and there is a direct edge, or an arrow, from vertex to (denoted as ), if and . Chain components of are obtained by removing all arrows in and taking the connectivity components of the remaining undirected graph. A chain graph with only undirected edges is known as an undirected graph (). A chain graph with only directed edges and without any directed cycles is known as a directed acyclic graph ().

If , then we call a parent of and a child of . In general, we use and to represent the collection of parents and children of , respectively. If , then we call a neighbour of . We use to represent the set of neighbours of vertex in . If there is an edge between the vertices and , then we say that and are joined or adjacent. The family of contains and its parents, which is denoted as . For a set , we can define similarly , , and . The boundary of is the set of neighbours and parents of , which is denoted as . What is more, the boundary of a set is defined as . In case that the underlying CG is clearly specified, the subscript is often ignored to simplify the notations.

If there is a sequence of the vertices , , with or , for all , then we call it a route and and the ends of the route. Furthermore if for all there is , we say the route is descending and use to denote a descending route from vertex to . If the vertices in a route are all distinct from each other, then we call the route a path. If there is a descending path from vertex to , then we call an ancestor of vertex and a descendant of vertex , which are denoted as and , respectively. Similarly, the ancestral set of a set is defined as , and the descendant set is defined as . Furthermore, we define . A vertex without any children in is called a terminal. Besides, if , we call a terminal set. We call a route a pseudocycle if it satisfies . Besides, it is called a cycle if it also satisfies and are distinct vertices. A cycle or pseudocycle is directed if it is descending and has for some . Finally, if graph does not contain any directed cycles or directed pseudocycles, we will call it a (chain graph).

A section of a route in means any maximum undirected subroute in , for example, with . We call and the two ends of the section. Besides, if there is in for some (or in for some ), then we call (or ) a head-terminal, or else we call it a tail-terminal. The head-to-head section in reference to route is the section which has two head-terminals, and a non-head-to-head section if it has one head-terminal at most. We use to denote the set of vertices of a section . If with , then we consider that section is outside of the set . If , then we consider that is hit by set .

Another important concept in is complex . A complex in is an especial path , , with , , and in , where there is no extra edge among vertices on the path. The vertices and are called the parents of complex , denoted as , and is called the region of . What is more, the complex with vertex as one of its parents is denoted by . At last, the Markov Blanket of vertex is defined as .

Next, we give the definition of -separation in . If a section of route satisfies the following two presentations, (1) if is a head-to-head section in reference to and is outside of set , or (2) is not a head-to-head section in reference to and is hit by set , then we say that is -separated by the set in . What is more, we say that the disjoint set in is -separated by the disjoint set if every route between and is -separated by , and form a -separation which is denoted by .

If two share the same -separation patterns over the same set of vertices, then we say that they are* Markov equivalent*. All chain graphs, which are Markov equivalent with each other, form equivalent class, which is known as* Markov equivalent class*. It is well known that two are* Markov equivalent* if and only if they share the identical global skeleton and complexes [20].

If a probability distribution over permits the following factorization [3], where is the collection of chain components of and after given , the conditional probability distribution of is denoted as ; then we say that is a compatible probability distribution in reference to . It is easy to check that if is a compatible probability distribution in reference to , we have where represents the conditional independence between and given in . If the condition is strengthened to then we say that is faithful to . Let be the family of compatible probability distributions in reference to . A chain graph model is defined as , where is a family of probability distributions that are compatible with .

*Example 1. *Consider a in Figure 1, where and , . We have , , , , , , , and . and are complexes, where and , respectively. The path between and is -separated by or , while the path between and is -separated by the empty set. The sets and are -separated by the set , and form a -separation .