• Bernoulli
  • Volume 21, Number 3 (2015), 1467-1493.

Standard imsets for undirected and chain graphical models

Takuya Kashimura and Akimichi Takemura

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We derive standard imsets for undirected graphical models and chain graphical models. Standard imsets for undirected graphical models are described in terms of minimal triangulations for maximal prime subgraphs of the undirected graphs. For describing standard imsets for chain graphical models, we first define a triangulation of a chain graph. We then use the triangulation to generalize our results for the undirected graphs to chain graphs.

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Bernoulli, Volume 21, Number 3 (2015), 1467-1493.

Received: February 2011
Revised: January 2014
First available in Project Euclid: 27 May 2015

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Zentralblatt MATH identifier

conditional independence decomposable graph maximal prime subgraph triangulation


Kashimura, Takuya; Takemura, Akimichi. Standard imsets for undirected and chain graphical models. Bernoulli 21 (2015), no. 3, 1467--1493. doi:10.3150/14-BEJ611.

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  • [1] Andersson, S.A., Madigan, D. and Perlman, M.D. (1997). On the Markov equivalence of chain graphs, undirected graphs, and acyclic digraphs. Scand. J. Stat. 24 81–102.
  • [2] Bouckaert, R., Hemmecke, R., Lindner, S. and Studený, M. (2010). Efficient algorithms for conditional independence inference. J. Mach. Learn. Res. 11 3453–3479.
  • [3] Frydenberg, M. (1990). The chain graph Markov property. Scand. J. Stat. 17 333–353.
  • [4] Geiger, D. and Pearl, J. (1993). Logical and algorithmic properties of conditional independence and graphical models. Ann. Statist. 21 2001–2021.
  • [5] Hara, H. and Takemura, A. (2010). A localization approach to improve iterative proportional scaling in Gaussian graphical models. Comm. Statist. Theory Methods 39 1643–1654.
  • [6] Heggernes, P. (2006). Minimal triangulations of graphs: A survey. Discrete Math. 306 297–317.
  • [7] Hemmecke, R., Lindner, S. and Studený, M. (2010). Learning restricted Bayesian network structures. Preprint. Available at arXiv:1011.6664v1.
  • [8] Hemmecke, R., Morton, J., Shiu, A., Sturmfels, B. and Wienand, O. (2008). Three counter-examples on semi-graphoids. Combin. Probab. Comput. 17 239–257.
  • [9] Lauritzen, S.L. (1996). Graphical Models. Oxford Statistical Science Series 17. New York: Oxford Univ. Press.
  • [10] Lauritzen, S.L., Dawid, A.P., Larsen, B.N. and Leimer, H.-G. (1990). Independence properties of directed Markov fields. Networks 20 491–505. Special issue on influence diagrams.
  • [11] Leimer, H.-G. (1993). Optimal decomposition by clique separators. Discrete Math. 113 99–123.
  • [12] Ohtsuki, T., Cheung, L.K. and Fujisawa, T. (1976). Minimal triangulation of a graph and optimal pivoting order in a sparse matrix. J. Math. Anal. Appl. 54 622–633.
  • [13] Pearl, J. (1988). Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. The Morgan Kaufmann Series in Representation and Reasoning. San Mateo, CA: Morgan Kaufmann.
  • [14] Studený, M. (1994/1995). Description of structures of stochastic conditional independence by means of faces and imsets (a series of three papers). Int. J. Gen. Syst. 23 123–137, 201–219, 323–341.
  • [15] Studený, M. (2001). On non-graphical description of models of conditional independence structure. In HSSS Workshop on Stochastic Systems for Individual Behaviours, Louvain la Neuve, Belgium.
  • [16] Studený, M. (2005). Probabilistic Conditional Independence Structures. London: Springer.
  • [17] Studený, M. and Bouckaert, R.R. (1998). On chain graph models for description of conditional independence structures. Ann. Statist. 26 1434–1495.
  • [18] Studený, M., Roverato, A. and Štěpánová, Š. (2009). Two operations of merging and splitting components in a chain graph. Kybernetika (Prague) 45 208–248.
  • [19] Studený, M. and Vomlel, J. (2009). A reconstruction algorithm for the essential graph. Internat. J. Approx. Reason. 50 385–413.
  • [20] Studený, M., Vomlel, J. and Hemmecke, R. (2010). A geometric view on learning Bayesian network structures. Internat. J. Approx. Reason. 51 573–586.
  • [21] Verma, T. and Pearl, J. (1990). Causal networks: Semantics and expressiveness. In Uncertainty in Artificial Intelligence, 4. Mach. Intelligence Pattern Recogn. 9 69–76. Amsterdam: North-Holland.
  • [22] Vomlel, J. and Studený, M. (2007). Graphical and algebraic representatives of conditional independence models. In Advances in Probabilistic Graphical Models 55–80. Berlin: Springer.