Geodesic
Two operations of merging and splitting components in a chain graph
Kybernetika, Tome 45 (2009) no. 2, pp. 208-248 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

In this paper we study two operations of merging components in a chain graph, which appear to be elementary operations yielding an equivalent graph in the respective sense. At first, we recall basic results on the operation of feasible merging components, which is related to classic LWF (Lauritzen, Wermuth and Frydenberg) Markov equivalence of chain graphs. These results are used to get a graphical characterisation of factorisation equivalence of classic chain graphs. As another example of the use of this operation, we derive some important invariants of LWF Markov equivalence of chain graphs. Last, we recall analogous basic results on the operation of legal merging components. This operation is related to the so-called strong equivalence of chain graphs, which includes both classic LWF equivalence and alternative AMP (Andersson, Madigan and Perlman) Markov equivalence.
In this paper we study two operations of merging components in a chain graph, which appear to be elementary operations yielding an equivalent graph in the respective sense. At first, we recall basic results on the operation of feasible merging components, which is related to classic LWF (Lauritzen, Wermuth and Frydenberg) Markov equivalence of chain graphs. These results are used to get a graphical characterisation of factorisation equivalence of classic chain graphs. As another example of the use of this operation, we derive some important invariants of LWF Markov equivalence of chain graphs. Last, we recall analogous basic results on the operation of legal merging components. This operation is related to the so-called strong equivalence of chain graphs, which includes both classic LWF equivalence and alternative AMP (Andersson, Madigan and Perlman) Markov equivalence.
Classification : 05C90, 62H05, 68T30
Keywords: chain graph; essential graph; factorisation equivalence; feasible merging components; legal merging components; strong equivalence 
@article{KYB_2009_45_2_a2,
     author = {Studen\'y, Milan and Roverato, Alberto and \v{S}t\v{e}p\'anov\'a, \v{S}\'arka},
     title = {Two operations of merging and splitting components in a chain graph},
     journal = {Kybernetika},
     pages = {208--248},
     year = {2009},
     volume = {45},
     number = {2},
     mrnumber = {2518149},
     zbl = {1252.62058},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/KYB_2009_45_2_a2/}
}
TY  - JOUR
AU  - Studený, Milan
AU  - Roverato, Alberto
AU  - Štěpánová, Šárka
TI  - Two operations of merging and splitting components in a chain graph
JO  - Kybernetika
PY  - 2009
SP  - 208
EP  - 248
VL  - 45
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/KYB_2009_45_2_a2/
LA  - en
ID  - KYB_2009_45_2_a2
ER  - 
%0 Journal Article
%A Studený, Milan
%A Roverato, Alberto
%A Štěpánová, Šárka
%T Two operations of merging and splitting components in a chain graph
%J Kybernetika
%D 2009
%P 208-248
%V 45
%N 2
%U http://geodesic.mathdoc.fr/item/KYB_2009_45_2_a2/
%G en
%F KYB_2009_45_2_a2
Studený, Milan; Roverato, Alberto; Štěpánová, Šárka. Two operations of merging and splitting components in a chain graph. Kybernetika, Tome 45 (2009) no. 2, pp. 208-248. http://geodesic.mathdoc.fr/item/KYB_2009_45_2_a2/

[1] S. A. Andersson, D. Madigan, and M. D. Perlman: An alternative Markov property for chain graphs. In: Uncertainty in Artificial Intelligence, Proc. Twelfth Conference (F. Jensen and E. Horvitz, eds.), Morgan Kaufmann, San Francisco 1996, pp. 40–48. | MR

[2] S. A. Andersson, D. Madigan, and M. D. Perlman: A characterization of Markov equivalence classes for acyclic digraphs. Ann. Statist. 25 (1997), 505–541. | MR

[3] S. A. Andersson, D. Madigan, and M. D. Perlman: On the Markov equivalence of chain graphs, undirected graphs and acyclic digraphs. Scand. J. Statist. 24 (1997), 81–102. | MR

[4] S. A. Andersson, D. Madigan, and M. D. Perlman: Alternative Markov properties for chain graphs. Scand. J. Statist. 28 (2001), 33–85. | MR

[5] D. M. Chickering: A transformational characterization of equivalent Bayesian network structures. In: Uncertainty in Artificial Intelligence, Proc. Eleventh Conference (P. Besnard and S. Hanks, eds.), Morgan Kaufmann, San Francisco 1995, pp. 87–98. | MR

[6] M. Frydenberg: The chain graph Markov property. Scand. J. Statist. 17 (1990), 333–353. | MR | Zbl

[7] S. L. Lauritzen and N. Wermuth: Mixed Interaction Models. Research Report No. R-84-8, Inst. Elec. Sys., University of Aalborg 1984.

[8] S. L. Lauritzen and N. Wermuth: Graphical models for association between variables, some of which are qualitative and some quantitative. Ann. Statist. 17 (1989), 31–57. | MR

[9] S. L. Lauritzen: Graphical Models. Clarendon Press, Oxford 1996. | MR | Zbl

[10] J. Pearl: Probabilistic Reasoning in Intelligent Systems. Morgan Kaufmann, San Mateo 1988. | MR | Zbl

[11] A. Roverato: A unified approach to the characterisation of equivalence classes of DAGs, chain graphs with no flags and chain graphs. Scand. J. Statist. 32 (2005), 295–312. | MR

[12] A. Roverato and M. Studený: A graphical representation of equivalence classes of AMP chain graphs. J. Machine Learning Research 7 (2006), 1045–1078. | MR

[13] Š. Štěpánová: Equivalence of Chain Graphs (in Czech). Diploma Thesis, Charles University, Prague 2003.

[14] M. Studený: A recovery algorithm for chain graphs. Internat. J. Approx. Reasoning 17 (1997), 265–293. | MR

[15] M. Studený: Characterization of essential graphs by means of the operation of legal merging of components. Internat. J. Uncertainty, Fuzziness and Knowledge-Based Systems 12 (2004), 43–62. | MR

[16] M. Studený and J. Vomlel: Transition between graphical and algebraic representatives of Bayesian network models. In: Proc. 2nd European Workshop on Probabilistic Graphical Models (P. Lucas ed.), Leiden 2004, pp. 193–200.

[17] M. Studený: Probabilistic Conditional Independence Structures. Springer-Verlag, London 2005.

[18] T. Verma and J. Pearl: Equivalence and synthesis of causal models. In: Uncertainty in Artificial Intelligence, Proc. Sixth Conference (P. Bonissone, M. Henrion, L. N. Kanal, and J. F. Lemmer, eds.), North-Holland, Amsterdam 1991, pp. 255–270.

[19] M. Volf and M. Studený: A graphical characterization of the largest chain graphs. Internat. J. Approx. Reasoning 20 (1999), 209–236. | MR