Test (2012) 21:253–254DOI 10.1007/s11749-012-0284-4

D I S C U S S I O N

Comments on: Sequences of regressionsand their independences

Robert Castelo

Published online: 3 April 2012© Sociedad de Estadística e Investigación Operativa 2012

I would like first to congratulate Nanny Wermuth and Kayvan Sadeghi for their com-prehensive and insightful review of regression graphs, which also takes the readerthrough some of the milestones in the theory of graphical Markov models. Amongthe many aspects involved in the thorough description of the concepts and propertiesof regression graphs provided by the authors, I would like to draw the attention ofthis commentary to their relationship with respect to other types of graphical Markovmodels.

Markov properties, such as the global Markov property on purely undirectedgraphs, provide the connection between graphs and probability distributions, whichallows one to derive intuitive interpretations from complex statistical models. Theextent of these interpretations depends on the type of graph employed to define thegraphical Markov model, and therefore, its topological properties tell us somethingabout the class of statistical models represented by the graph. A canonical exampleare concentration graphs Markov equivalent to acyclic digraphs (DAGs), which cor-respond to chordal graphs (Wermuth 1980; Kiiveri et al. 1984). As pointed out inthe paper, chordal graphs permit one to employ efficient fitting procedures to esti-mate structure and parameters from data, while chordless cycles on more than threevertices occurring in concentration graphs impose the requirement of iterative fittingalgorithms for that purpose. Analogous findings on computational advantages con-ferred by chordal graphs have been also exploited in the field of databases (Beeri etal. 1983) in computer science.

Communicated by Domingo Morales.

This comment refers to the invited paper available at doi:10.1007/s11749-012-0290-6.

R. Castelo (�)Department of Experimental and Health Sciences, Universitat Pompeu Fabra, Barcelona, Spaine-mail: robert.castelo@upf.edu

254 R. Castelo

This picture may suggest that different types of graph lead to a mosaic of distinctisolated classes of graphical Markov models. However, Wermuth and Sadeghi in thispaper rightly show that by inserting edges in a regression graph it can be convertedinto another graphical Markov model of one of the intersecting classes, suggesting infact that all the distinct classes of graphical Markov models are interlaced. This ob-servation was already made by Andersson et al. (1995, p. 38) in the context of latticeconditional independence (LCI) models, which coincide with the class of transitiveDAG models (DAGs without transition Vs), where they show that every LCI modelincludes and it is included in at least one DAG model. Those authors also concludedthat every conditional independence restriction is equivalent to a simple LCI modeland thus any graphical Markov model could be described by the intersection of allLCI models that contain it. This result was later expanded with the characterization ofthe class of DAGs with exclusively source Vs, also known as tree conditional indepen-dence (TCI) models (Castelo and Siebes 2003), whose Markov equivalence classesare represented by P4-free chordal graphs, as recalled also in this paper by Wermuthand Sadeghi. In this latter restricted subclass of chordal graphs the intersection ofall cliques is always non-empty in each connected component (Castelo and Wormald2003) and one such connected graph constitutes the simplest and most constraintgraphical representation of one single conditional independence restriction.

The interlaced structure of graphical Markov models can provide a solution to theproblem of selecting a regression graph from data within the vast search space of suchmodels by searching first within a model class with fast data fitting procedures andthen refine that model to select the final regression graph. While the authors claimthat LWF chain graphs may constitute one such class of models because model fit-ting is possible in terms of single response regressions and Markov equivalence canbe handled in the same way as in DAGs, the concept of vertex separator is much morecomplex. This contrasts with undirected graphs where this concept is much simplerand thus more amenable for working with marginal distributions of size (q + 2) < n

whenever the number of random variables p is larger than the number of observa-tions n (Castelo and Roverato 2006).

References

Andersson S, Madigan D, Perlman M, Triggs C (1995) On the relation between conditional independencemodels determined by finite distributive lattices and by directed acyclic graphs. J Stat Plan Inference48:25–46

Beeri C, Fagin R, Maier D, Yannakakis M (1983) On the desirability of acyclic database schemes. J ACM30(3):479–513

Castelo R, Roverato A (2006) A robust procedure for Gaussian graphical model search from microarraydata with p larger than n. J Mach Learn Res 7:2621–2650

Castelo R, Siebes A (2003) A characterization of moral transitive acyclic directed graph Markov modelsas labeled trees. J Stat Plan Inference 115:235–259

Castelo R, Wormald N (2003) Enumeration of P4-free chordal graphs. Graphs Comb 19:467–474Kiiveri H, Speed T, Carlin J (1984) Recursive causal models. J Aust Math Soc A 36:30–52Wermuth N (1980) Linear recursive equations, covariance selection, and path analysis. J Am Stat Assoc

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