In this paper we discuss four problems regarding Markov equivalences for subclasses of loopless mixed graphs. We classify these four problems as finding conditions for internal Markov equivalence, which is Markov equivalence within a subclass, for external Markov equivalence, which is Markov equivalence between subclasses, for representational Markov equivalence, which is the possibility of a graph from a subclass being Markov equivalent to a graph from another subclass, and finding algorithms to generate a graph from a certain subclass that is Markov equivalent to a given graph. We particularly focus on the class of maximal ancestral graphs and its subclasses, namely regression graphs, bidirected graphs, undirected graphs, and directed acyclic graphs, and present novel results for representational Markov equivalence and algorithms.
Introduction and motivation. In graphical Markov models several classes of graphs have been used in recent years. A common feature of all these graphs is that their nodes correspond to random variables, and they represent conditional independence statements of the node set by specific interpretations of missing edges.
These graphs contain up to three different types of edges. Sadeghi & Lauritzen (2011) gathered most classes of graphs defined in the literature under a unifying class of loopless mixed graphs (LMGs). These contain Summary graphs (SGs) (Wermuth, 2011), (maximal) ancestral graphs (MAGs) (Richardson & Spirtes, 2002), acyclic directed mixed graphs (ADMGs) (Spirtes et al., 1997), regression chain graphs (RCGs) (Cox & Wermuth, 1993;Wermuth & Cox, 2004;Wermuth & Sadeghi, 2011), undirected or concentration graphs (UGs) (Darroch et al., 1980;Lauritzen, 1996), bidirected or covariance graphs (BGs) (Cox & Wermuth, 1993;Wermuth & Cox, 1998), and directed acyclic graphs (DAGs) (Kiiveri et al., 1984;Lauritzen, 1996).
For the above graphs, in general, two graphs of different types or even two graphs of the same type may induce the same independencies. Such graphs are said to be Markov equivalent. It is important to explore the similar characteristics of Markov equivalent graphs, and to find the ways of generating graphs of a certain type with the same independence structure from a given graph. Some questions for Markov equivalences. There are four main questions regarding Markov equivalence for different types of graphs:
Internal Markov equivalence: The first natural question that arises in this context is regarding when two graphs of the same type are Markov equivalent. This question may be answered for DAGs, MAGs, or other subclasses of LMGs.
External Markov equivalence: In addition to Markov equivalence for graphs of the same type, one can discuss Markov equivalence between two graphs of different types.
Representational Markov equivalence: Before checking external Markov equivalence, however, it is essential to check whether and under what conditions a graph of a certain type can be Markov equivalent to a graph of another type.
One can also present some algorithms to generate a graph of a certain type that is Markov equivalent to a given graph of a different type.
In this paper we gather and simplify the existing results in the literature for internal and external Markov equivalences, and give novel results for representational Markov equivalence and algorithms.
Some earlier results on Markov equivalence for graphs. Results concerning Markov equivalence for different classes of graphs have been obtained independently in the statistical literature on specifying types of multivariate statistical models, and in the computer science literature on deciding on special properties of a given graph or on designing fast algorithms for transforming graphs. In the literature on graphical Markov models two of the early results concerning Markov equivalence for DAGs and chain graphs were respectively given in Verma & Pearl (1990) and Frydenberg (1990). Two of the later results by Zhao et al. (2004) and Ali et al. (2009) respectively provided theoretically neat and computationally fast conditions for Markov equivalence for maximal ancestral graphs.
Besides these, Pearl & Wermuth (1994) provided conditions for Markov equivalence for bidirected graphs and DAGs. Spirtes & Richardson (1997) gave some conditions for Markov equivalence for maximal ancestral graphs, in which the polynomial computational complexity claim was wrong.
Efficient algorithms for deciding whether a UG can be oriented into a DAG became available in the computer science literature under the name of perfect elimination orientations; see Tarjan & Yannakakis (1984), whose algorithm can be run in O(|V | + |E|).
Another such linear algorithm can be found in Blair & Peyton (1992). An algorithm for generating a Markov equivalent DAG from a bidirected graph is the special case of the algorithm given in Zhao et al. (2004).
Structure of the paper. In the next section we define the unifying class of LMGs, and provide some basic graph theoretical definitions needed for our results.
In Section 3 we present the subclasses of LMGs, and we formally define the subclasses of interest in this paper. We also define a so-called separation criterion, called m-separation, to provide an interpretation of independencies for the graphs.
In Section 4 we formally define Markov equivalence, define maximality and explain its importance for Markov equivalences, and motivate why we consider Markov equivalence for the class of MAGs.
In Section 5 we gather the conditions existing in the literature for internal Markov equivalence for the class of MAGs and its subclasses, and give conditions for their external Markov equivalence.
In Section 6 we discuss the conditions for representational Markov equivalence for MAGs and its subclasses to a specific subclass, and
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