S. A. Andersson, D. Madigan and M. D. Perlman. A characterization of Markov equivalence classes for acyclic digraphs. The Annals of Statistics, 25:505--541, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....class would seem rather ine#cient (since they inadvertedly keep proposing the same model) We conclude by briefly providing some preliminary insights on this matter. It turns out that equivalence classes can be compactly represented by (certain class of) partially directed acyclic graphs or PDAGs [1, 5]. PDAGs include directed as well as undirected arcs. Chickering [5] provides an algorithm that takes a given DAG G and outputs the PDAG G that uniquely represents its equivalence class [G] Since G and G have the same connectivity pattern (ig LVFAILURE HISTORY LVEDVOLUME STROKEVOLUME ....

S.A. Andersson, D. Madigan, and M.D. Perlman. A characterization of markov equivalence classes for acyclic digraphs. Annals of Statistics, 25:505--541, 1997.

....directed and the process continues. Verma and Pearl (1992) present an algorithm of this type that is known to be sound, but not complete. That is, every directed edge in the final pdag is provably compelled, but not every undirected edge is provably reversible. More recently, both Meek (1995) and Anderson et al. 1995) havederived sound and complete algorithms for constructing a completed pdag representation. The algorithm we present in this section takes a dag as input, and labels every edge in the dag as either compelled or reversible. We show that the algorithm is correct and discuss an implementation that ....

Anderson, S. A., Madigan, D., and Perlman, M. D. (1995). A characterization of markov equivalence classes for acyclic digraphs. Technical Report 287, UniversityofWashington, Department of Statistics.

....[17, 9] generalized the concept of moral graph and introduced a moralization criterion for reading independency statements from a CG. Frydenberg [9] characterized CGs with the same Markov ON CHAIN GRAPH MODELS 3 property (that is producing the same CG model) and Andersson, Madigan and Perlman [3] used special CGs to represent uniquely classes of Markov equivalent DAGs. Whittaker [31] in his book gave several examples of the use of CGs, and other recent works also deal with them [6, 20, 23, 30] the most comprehensive account is provided by the book [19] Several results proved here were ....

.... graph (it has the greatest number of lines within the class of Markov equivalent CGs) Note that this result has no analogy in the case of DAGs: Markov equivalent DAGs have no distinguished members and have to be represented by more general mathematical objects, like for example, essential graphs [3]. This fact causes many complications for search procedures within the class of DAG models (that is, the procedures for estimation of the most suitable DAG model approximation on the basis of statistical data) However, in the case of CGs one can simply represent every CG model by the ....

Andersson, S. A., Madigan, D. and Perlman, M. D. A characterization of Markov equivalence classes for acyclic digraphs. Ann. Statist. 25 (1997) 505--541. 62 M. STUDEN ' Y AND R. R. BOUCKAERT

....represented in form of a junction tree. This is a basis of a local computation algorithm based on probability propagation see (Lauritzen 1996) Perhaps one can somehow to utilize the fact that in the formula from 4. 1 every P Cjpa(C) or equivalently P C[pa(C) complies with local decomposable model. Remark Andersson et al. 1997) proposed to represent every class D of Markov equivalent Bayesian networks by the essential graph of D which has the same underlying graph as every D 2 D and only those arrows which are shared by all graphs in D. This is always a chain graph. Example 4.2 shows that the corresponding Markov ....

S.A. Andersson, D. Madigan, and M.D. Perlman (1997). A characterization of Markov equivalence classes for acyclic digraphs. Ann. Statist. 25: 505541.

....[9] Several recent works suggest that CGs attract attention of researchers. Whittaker [23] gave several examples of use of CGs, Cox and Wermuth [6] considered a wider class of chain graphs having further two types of edges, namely dashed lines and dashed arrows . Andersson, Madigan and Perlman [1] used special CGs, called essential graphs, to represent uniquely classes of Markov equivalent DAGs and characterized the essential graphs in graphical terms. Meek [16] followed the method from [22] and proposed an algorithm which on basis of the conditional independence structure given by a DAG ....

....by some criterion for CGs, for example by moralization criterion. In fact, the concept of essential graph also provides a solution of the mentioned problem because of it also belongs to the class of CGs which are Markov equivalent to the considered DAG and one can use the criterion for CGs see [1]. The second remark concerns the pattern recovery algorithm. It has an important feature: it depends only on predicates hu; vj Gammai and hu; vj wi introduced in Definition 3.2. Especially, two CG models which coincide on these predicates must be equal The number of such predicates is polynomial ....

Andersson, S.A., Madigan, D., and Perlman, M.D., A characterization of Markov equivalence classes for acyclic digraphs, tech. rep. n. 287, University of Washington, Seattle, 1995, submitted to Ann. Statist.

....Figure 10 A pattern has the same adjacencies as the path diagrams in the covariance equivalence class that it represents. In addition, an edge is oriented as X Z in the pattern if and only if it is oriented as X Z in every path diagram in the simple covariance equivalence class. Meek 1995, Andersson et al. 1995, and Chickering 1995 show how to generate a pattern from an acyclic graph in O(E) time (where E is the number of edges. In the case of acyclic path diagrams which may also contain latent variables, and the case of cyclic path diagrams which do not contain latent variables, there is an object ....

Andersson, S., Madigan, D., and Perlman, M. (1995) A Characterization of Markov Equivalence Classes for Acyclic Digraphs, Technical Report 287, Department of Statistics, University of Washington.

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S. A. Andersson, D. Madigan and M. D. Perlman. A characterization of Markov equivalence classes for acyclic digraphs. The Annals of Statistics, 25:505--541, 1997.

....of the ADG is complete, then the equivalence class contains exactly n ADGs. Model search and selection algorithms that ignore Markov equivalence may therefore be extremely inefficient. Treating each Markov equivalence class as a single model would overcome these difficulties. An essential graph [1] is a graphical representation of a Markov equivalence class that can have both directed and undirected edges. Each directed edge in the essential graph is oriented in the same direction in all members of the equivalence class. Each of the undirected edges will appear in different directions in at ....

....classes or for their ratio r n to the number of ADGs. Robinson [25] found a recursive formula for the number of ADGs on n vertices; thus, given either of the two unknown quantities above the other can be computed. The enumerative question has been partially addressed by two independent researchers [1, 34]. Both found the number of equivalence classes for n 5 vertices via a combination of manual and computer methods. Neither study was able to make any predictions about the values for n 6. Steinsky [30] found a recursive formula for the number of equivalence classes of size 1 only, and using a ....

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S.A. Andersson, D. Madigan, M.D. Perlman, A characterization of Markov equivalence classes for acyclic digraphs, Annals of Statistics 25 (1997) 505-541.

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Andersson, S. A., D. Madigan, and M. D. Perlman (1995b). A characterization of Markov equivalence classes for acyclic digraphs.

....Intelligence community focuses much of its effort on ADG models. Lattice conditional independence (LCI) models were introduced by Andersson and Perlman (1993) under the additional assumption of multivariate normality, motivated by analogy with the lattice structure of balanced ANOVA designs (cf. Andersson (1990)) In an LCI model, formally defined in Sections 4 and 5, conditional independence relations among the variables are specified by the intersection properties of a finite distributive lattice (cf. Figure 1.2) LCI models share the desirable statistical properties of ADG models: they admit tractable ....

Andersson, S. A., D. Madigan, and M. D. Perlman (1995b). A characterization of Markov equivalence classes for acyclic digraphs. Submitted for publication.

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S. Andersson, D. Madigan, and M. Perlman. A Characterization of Markov equivalence classes for acyclic digraphs. Annals of Statistics, 25:505--541, 1997.

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Andersson, S. A., D. Madigan, and M. D. Perlman (1997). A characterization of Markov equivalence classes for acyclic digraphs. Annals of Statistics 25 (2), 505--541.

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S. A. Andersson, D. Madigan, and M. D. Perlman. A characterization of Markov equivalence classes for acyclic digraphs. Annals of Statistics, 25(2):505--541, 1997.

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S. A. Anderson, D. Madigan, M. D. Perlman, A characterization of markov equivalence classes for acyclic digraphs, Annals of Statistics 25 (1997) 505--541.

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S.A. Andersson, D. Madigan, and M.D. Perlman (1995). A characterization of Markov equivalence classes for acyclic digraphs. Tech. rep. 287, University of Washington, Seattle, submitted to Ann. Statist.

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