Markov Equivalence Research Articles

A Graphical Representation of Equivalence Classes of AMP Chain Graphs

This paper deals with chain graph models under alternative AMP interpretation. A new representative of an AMP Markov equivalence class, called the largest deflagged graph, is proposed. The represen...

Algebraic Markov equivalence for links in three-manifolds

Let B-n denote the classical braid group on n strands and let the mixed braid group B-m,B-n be the subgroup of Bm+n comprising braids for which the first m strands form the identity braid. Let B-m,B-infinity = boolean OR(n) B-m,B-n. We describe explicit algebraic moves on B-m,B-infinity such that equivalence classes under these moves classify oriented links up to isotopy in a link complement or in a closed, connected, oriented three-manifold. The moves depend on a fixed link representing the manifold in S-3. More precisely, for link complements the moves are the two familiar moves of the classical Markov equivalence together with 'twisted' conjugation by certain loops a(i). This means premultiplication by a(i)(-1) and postmultiplication by a 'combed' version of a(i). For closed three-manifolds there is an additional set of 'combed' band moves that correspond to sliding moves over the surgery link. The main tool in the proofs is the one-move Markov theorem using L-moves (adding in-box crossings). The resulting algebraic classification is a direct extension of the classical Markov theorem that classifies links in S-3 up to isotopy, and potentially leads to powerful new link invariants, which have been explored in special cases by the first author. It also provides a controlled range of isotopy moves, useful for studying skein modules of three-manifolds.

Characterizing Markov equivalence classes for AMP chain graph models

Chain graphs (CG) (= adicyclic graphs) use undirected and directed edges to represent both structural and associative dependences. Like acyclic directed graphs (ADGs), the CG associated with a statistical Markov model may not be unique, so CGs fall into Markov equivalence classes, which may be superexponentially large, leading to unidentifiability and computational inefficiency in model search and selection. It is shown here that, under the Andersson-Madigan-Perlman (AMP) interpretation of a CG, each Markov-equivalence class can be uniquely represented by a single distinguished CG, the AMP essential graph, that is itself simultaneously Markov equivalent to all CGs in the AMP Markov equivalence class. A complete characterization of AMP essential graphs is obtained. Like the essential graph previously introduced for ADGs, the AMP essential graph will play a fundamental role for inference and model search and selection for AMP CG models.

On Block Ordering of Variables in Graphical Modelling

In graphical modelling, the existence of substantive background knowledge on block ordering of variables is used to perform structural learning within the family of chain graphs in which every block corresponds to an undirected graph and edges joining vertices in different blocks are directed in accordance with the ordering. We show that this practice may lead to an inappropriate restriction of the search space and introduce the concept of labelled block ordering B corresponding to a family of B-consistent chain graphs in which every block may be either an undirected graph or a directed acyclic graph or, more generally, a chain graph. In this way we provide a flexible tool for specifying subsets of chain graphs, and we observe that the most relevant subsets of chain graphs considered in the literature are families of B-consistent chain graphs for the appropriate choice of B. Structural learning within a family of B-consistent chain graphs requires to deal with Markov equivalence. We provide a graphical characterisation of equivalence classes of B-consistent chain graphs, namely the B-essential graphs, as well as a procedure to construct the B-essential graph for any given equivalence class of B-consistent chain graphs. Both largest chain graphs and essential graphs turn out to be special cases of B-essential graphs.

A Unified Approach to the Characterization of Equivalence Classes of DAGs, Chain Graphs with no Flags and Chain Graphs

. A Markov property associates a set of conditional independencies to a graph. Two alternative Markov properties are available for chain graphs (CGs), the Lauritzen–Wermuth–Frydenberg (LWF) and the Andersson–Madigan– Perlman (AMP) Markov properties, which are different in general but coincide for the subclass of CGs with no flags. Markov equivalence induces a partition of the class of CGs into equivalence classes and every equivalence class contains a, possibly empty, subclass of CGs with no flags itself containing a, possibly empty, subclass of directed acyclic graphs (DAGs). LWF-Markov equivalence classes of CGs can be naturally characterized by means of the so-called largest CGs, whereas a graphical characterization of equivalence classes of DAGs is provided by the essential graphs. In this paper, we show the existence of largest CGs with no flags that provide a natural characterization of equivalence classes of CGs of this kind, with respect to both the LWF- and the AMP-Markov properties. We propose a procedure for the construction of the largest CGs, the largest CGs with no flags and the essential graphs, thereby providing a unified approach to the problem. As by-products we obtain a characterization of graphs that are largest CGs with no flags and an alternative characterization of graphs which are largest CGs. Furthermore, a known characterization of the essential graphs is shown to be a special case of our more general framework. The three graphical characterizations have a common structure: they use two versions of a locally verifiable graphical rule. Moreover, in case of DAGs, an immediate comparison of three characterizing graphs is possible.

An efficient algorithm for finding the largest chain graph according to a given chain graph

Chain graph (CG) is a general model of graphical Markov models. Some different chain graphs may describe the same conditional independence structure, then we say that these CGs are Markov equivalent. In 1990 Frydenberg showed that every class of Markov equivalent CGs has a CG which is called the largest chain graph with the greatest number of lines. This paper presents an efficient algorithm for finding the largest chain graph of the corresponding Markov equivalent class of a given CG. The computational complexity of the algorithm is O ( n 3 ). It is more efficient than the complexity O ( n !) of the present algorithms. Also a more intuitive graphical characterization of the largest chain graph is provided based on the algorithm in this paper.

Formulas for counting acyclic digraph Markov equivalence classes

Multivariate Markov dependencies between different variables often can be represented graphically using acyclic digraphs (ADGs). In certain cases, though, different ADGs represent the same statistical model, thus leading to a set of equivalence classes of ADGs that constitute the true universe of available graphical models. Building upon the previously known formulas for counting the number of acyclic digraphs and the number of equivalence classes of size 1, formulas are developed to count ADG equivalence classes of arbitrary size, based on the chordal graph configurations that produce a class of that size. Theorems to validate the formulas as well as to aid in determining the appropriate chordal graphs to use for a given class size are included.

Enumeration of labelled chain graphs and labelled essential directed acyclic graphs

A chain graph is a digraph whose strong components are undirected graphs and a directed acyclic graph (ADG or DAG) G is essential if the Markov equivalence class of G consists of only one element. We provide recurrence relations for counting labelled chain graphs by the number of chain components and vertices; labelled essential DAGs by the number of vertices. The second one is a lower bound for the number of labelled essential graphs. The formula for labelled chain graphs can be extended in such a way, that allows us to count digraphs with two additional properties, which essential graphs have.

The size distribution for Markov equivalence classes of acyclic digraph models

Bayesian networks, equivalently graphical Markov models determined by acyclic digraphs or ADGs (also called directed acyclic graphs or dags), have proved to be both effective and efficient for representing complex multivariate dependence structures in terms of local relations. However, model search and selection is potentially complicated by the many-to-one correspondence between ADGs and the statistical models that they represent. If the ADGs/models ratio is large, search procedures based on unique graphical representations of equivalence classes of ADGs could provide substantial computational efficiency. Hitherto, the value of the ADGs/models ratio has been calculated only for graphs with n=5 or fewer vertices. In the present study, a computer program was written to enumerate the equivalence classes of ADG models and study the distributions of class sizes and number of edges for graphs up to n=10 vertices. The ratio of ADGs to numbers of classes appears to approach an asymptote of about 3.7. Distributions of the classes according to number of edges and class size were produced which also appear to be approaching asymptotic limits. Imposing a bound on the maximum number of parents to any vertex causes little change if the bound is sufficiently large, with four being a possible minimum. The program also includes a new variation of orderly algorithm for generating undirected graphs.

Alternative Markov Properties for Chain Graphs

Graphical Markov models use graphs to represent possible dependences among statistical variables. Lauritzen, Wermuth, and Frydenberg (LWF) introduced a Markov property for chain graphs (CG): graphs that can be used to represent both structural and associative dependences simultaneously and that include both undirected graphs (UG) and acyclic directed graphs (ADG) as special cases. Here an alternative Markov property (AMP) for CGs is introduced and shown to be the Markov property satisfied by a block‐recursive linear system with multivariate normal errors. This model can be decomposed into a collection of conditional normal models, each of which combines the features of multivariate linear regression models and covariance selection models, facilitating the estimation of its parameters. In the general case, necessary and sufficient conditions are given for the equivalence of the LWF and AMP Markov properties of a CG, for the AMP Markov equivalence of two CGs, for the AMP Markov equivalence of a CG to some ADG or decomposable UG, and for other equivalences. For CGs, in some ways the AMP property is a more direct extension of the ADG Markov property than is the LWF property.

A characterization of Markov equivalence for directed cyclic graphs

The concept of d-separation (Pearl, 1988) was originally defined for acyclic directed graphs, but there is a natural extension of the concept to directed graphs with cycles. When exactly the same set of d-separation relations hold in two directed graphs, no matter whether respectively cyclic or acyclic, we say that they are markov equivalent. In other words, when two directed cyclic graphs are Markov equivalent, the set of distributions that satisfy a natural extension of the global directed Markov condition (Lauritzen et al., 1990) is exactly the same for each graph. There is an obvious exponential (in the number of vertices) time algorithm for deciding Markov equivalence of two directed cyclic graphs: simply check all of the d-separation relations in each graph. In this paper I prove a theorem that gives necessary and sufficient conditions for two directed cyclic graphs to be Markov equivalent, where each of the conditions can be checked in polynomial time. Hence, the theorem can be easily adapted into a polynomial time algorithm for deciding the Markov equivalence of two directed cyclic graphs (Richardson, 1996).

A characterization of Markov equivalence classes for acyclic digraphs

Undirected graphs and acyclic digraphs (ADGs), as well as their mutual extension to chain graphs, are widely used to describe dependencies among variables in multivariate distributions. In particular, the likelihood functions of ADG models admit convenient recursive factorizations that often allow explicit maximum likelihood estimates and that are well suited to building Bayesian networks for expert systems. Whereas the undirected graph associated with a dependence model is uniquely determined, there may, however, be many ADGs that determine the same dependence (= Markov) model. Thus, the family of all ADGs with a given set of vertices is naturally partitioned into Markov-equivalence classes, each class being associated with a unique statistical model. Statistical procedures, such as model selection or model averaging, that fail to take into account these equivalence classes, may incur substantial computational or other inefficiencies. Here it is shown that each Markov-equivalence class is uniquely determined by a single chain graph, the essential graph, that is itself simultaneously Markov equivalent to all ADGs in the equivalence class. Essential graphs are characterized, a polynomial-time algorithm for their construction is given, and their applications to model selection and other statistical questions are described.

On the Markov Equivalence of Chain Graphs, Undirected Graphs, and Acyclic Digraphs

Graphical Markov models use undirected graphs (UDGs), acyclic directed graphs (ADGs), or (mixed) chain graphs to represent possible dependencies among random variables in a multivariate distribution. Whereas a UDG is uniquely determined by its associated Markov model, this is not true for ADGs or for general chain graphs (which include both UDGs and ADGs as special cases). This paper addresses three questions regarding the equivalence of graphical Markov models: when is a given chain graph Markov equivalent (1) to some UDG? (2) to some (at least one) ADG? (3) to some decomposable UDG? The answers are obtained by means of an extension of Frydenberg’s (1990) elegant graph‐theoretic characterization of the Markov equivalence of chain graphs.

On the Markov Equivalence of Chain Graphs, Undirected Graphs, and Acyclic Digraphs

On the Markov Equivalence of Chain Graphs, Undirected Graphs, and Acyclic Digraphs

Bayesian model averaging and model selection for markov equivalence classes of acyclic digraphs

Acyclic digraphs (ADGs) are widely used to describe dependences among variables in multivariate distributions. In particular, the likelihood functions of ADG models admit convenient recursive factorizations that often allow explicit maximum likelihood estimates and that are well suited to building Bayesian networks for expert systems. There may, however, be many ADGs that determine the same dependence (= Markov) model. Thus, the family of all ADGs with a given set of vertices is naturally partitioned into Markov-equivalence classes, each class being associated with a unique statistical model. Statistical procedures, such as model selection or model averaging, that fail to take into account these equivalence classes, may incur substantial computational or other inefficiencies. Recent results have shown that each Markov-equivalence class is uniquely determined by a single chain graph, the essential graph, that is itself Markov-equivalent simultaneously to all ADGs in the equivalence class. Here we propose t...

Markov measures determine the zeta function

AbstractWith the purpose of understanding when two subshifts of finite type are equivalent from the point of view of their spaces of Markov measures we propose the notion of Markov equivalence. We show that a Markov equivalence must respect the cycles (periodic orbits) of the subshifts. In particular, Markov equivalent subshifts of finite type have the same zeta function.

Inference for Discrete Markov Fields: The Simplest Nontrivial Case

Abstract Markov fields provide a general context for describing the strength and structure of spatial interactions. The Gibbs—Markov equivalence theorem (Preston 1974) parameterizes Markov fields via their neighborhood structures, yielding exponential families in canonical form. Likelihood inference is, therefore, apparently straightforward. The requisite normalizing constants, however, are obstreperous. Even when asymptotic characterizations can be obtained, substantial location errors arise during implementation. Moreover, Markov fields can exhibit phase transitions and long-range interactions, thereby creating identifiability problems. These issues are illustrated in the simplest nontrivial case—the classical Ising model of ferromagnetism.