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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Finding the minimal set for collapsible graphical models


Authors: Xiaofei Wang, Jianhua Guo and Xuming He
Journal: Proc. Amer. Math. Soc. 139 (2011), 361-373
MSC (2000): Primary 62-09, 05E05; Secondary 05C85
Published electronically: July 21, 2010
MathSciNet review: 2729097
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Abstract: A graphical model is said to be collapsible onto a set of variables if the implied model for the marginal distribution of those variables is the same as that given by the induced subgraph. We discuss the notion of collapsibility under multinomial, Gaussian, and mixed graphical models for undirected graphs, and we show that there exists a unique minimal set of variables onto which a graphical model can be collapsed. We also provide a useful algorithm for finding the minimal set and give examples to illustrate the utility of using collapsibility.


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Additional Information

Xiaofei Wang
Affiliation: Key Laboratory for Applied Statistics of MOE and School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, Jilin Province, People’s Republic of China
Email: mathswangxiaofei@yahoo.com.cn

Jianhua Guo
Affiliation: Key Laboratory for Applied Statistics of MOE and School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, Jilin Province, People’s Republic of China
Email: jhguo@nenu.edu.cn

Xuming He
Affiliation: Department of Statistics, University of Illinois, 725 S. Wright Street, Champaign, Illinois 61820
Email: x-he@uiuc.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10509-9
PII: S 0002-9939(2010)10509-9
Keywords: Collapsibility, decomposition, graphical models
Received by editor(s): April 4, 2009
Received by editor(s) in revised form: March 19, 2010
Published electronically: July 21, 2010
Additional Notes: This research was supported by the National Natural Science Foundation of China (Grants No. 10701022, 10871038, 10828102 and 10926186), the National 973 Key Project of China (2007CB311002), and the U.S. National Science Foundation award DMS-0630950
Communicated by: Edward C. Waymire
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.