Bernoulli
- Bernoulli
- Volume 16, Number 1 (2010), 208-233.
Minimal and minimal invariant Markov bases of decomposable models for contingency tables
Hisayuki Hara, Satoshi Aoki, and Akimichi Takemura
Full-text: Open access
Abstract
We study Markov bases of decomposable graphical models consisting of primitive moves (i.e., square-free moves of degree two) by determining the structure of fibers of sample size two. We show that the number of elements of fibers of sample size two are powers of two and we characterize primitive moves in Markov bases in terms of connected components of induced subgraphs of the independence graph of a hierarchical model. This allows us to derive a complete description of minimal Markov bases and minimal invariant Markov bases for decomposable models.
Article information
Source
Bernoulli, Volume 16, Number 1 (2010), 208-233.
Dates
First available in Project Euclid: 12 February 2010
Permanent link to this document
https://projecteuclid.org/euclid.bj/1265984709
Digital Object Identifier
doi:10.3150/09-BEJ207
Mathematical Reviews number (MathSciNet)
MR2648755
Zentralblatt MATH identifier
1203.62110
Keywords
chordal graph Gröbner bases independence graph invariance minimality symmetric group
Citation
Hara, Hisayuki; Aoki, Satoshi; Takemura, Akimichi. Minimal and minimal invariant Markov bases of decomposable models for contingency tables. Bernoulli 16 (2010), no. 1, 208--233. doi:10.3150/09-BEJ207. https://projecteuclid.org/euclid.bj/1265984709
References
- Agresti, A. (2002). Categorical Data Analysis, 2nd edn. New York: Wiley.
- Aoki, S. and Takemura, A. (2003). Minimal basis for a connected Markov chain over 3×3×K contingency tables with fixed two-dimensional marginals. Aust. N. Z. J. Stat. 45 229–249.
- Aoki, S. and Takemura, A. (2005). Markov chain Monte Carlo exact tests for incomplete two-way contingency table. J. Stat. Comput. Simul. 75 787–812.Mathematical Reviews (MathSciNet): MR2191323
Zentralblatt MATH: 1076.62060
Digital Object Identifier: doi:10.1080/00949650410001690079 - Aoki, S. and Takemura, A. (2008a). Minimal invariant Markov basis for sampling contingency tables with fixed marginals. Ann. Inst. Statist. Math. 60 229–256.Mathematical Reviews (MathSciNet): MR2403518
Zentralblatt MATH: 05560796
Digital Object Identifier: doi:10.1007/s10463-006-0089-x - Aoki, S. and Takemura, A. (2008b). The largest group of invariance for Markov bases and toric ideals. J. Symbolic Comput. 43 342–358.
- Aoki, S. and Takemura, A. (2009). Markov basis for design of experiments with three-level factors. In Algebraic and Geometric Methods in Statistics (P. Gibilisco, E. Riccomagno, M.P. Rogantin and H.P. Wynn, eds.). Cambridge Univ. Press. To appear.Mathematical Reviews (MathSciNet): MR2642668
- Aoki, S., Takemura, A. and Yoshida, R. (2008). Indispensable monomials of toric ideals and Markov bases. J. Symbolic Comput. 43 490–507.
- Basturk, R. (2008). Applying the many-facet Rasch model to evaluate PowerPoint presentation performance in higher education. Assessment & Evaluation in Higher Education 33 431–444.
- Chen, Y. and Small, D. (2005). Exact test for the Rasch model via sequential importance sampling. Psychometrika 70 11–30.Mathematical Reviews (MathSciNet): MR2272472
Digital Object Identifier: doi:10.1007/s11336-003-1069-1
Zentralblatt MATH: 1306.62395 - Diaconis, P. and Sturmfels, B. (1998). Algebraic algorithms for sampling from conditional distributions. Ann. Statist. 26 363–397.Mathematical Reviews (MathSciNet): MR1608156
Zentralblatt MATH: 0952.62088
Digital Object Identifier: doi:10.1214/aos/1030563990
Project Euclid: euclid.aos/1030563990 - Dobra, A. (2003). Markov bases for decomposable graphical models. Bernoulli 9 1093–1108.Mathematical Reviews (MathSciNet): MR2046819
Digital Object Identifier: doi:10.3150/bj/1072215202
Project Euclid: euclid.bj/1072215202
Zentralblatt MATH: 1053.62072 - Dobra, A. and Sullivant, S. (2004). A divide-and-conquer algorithm for generating Markov bases of multiway tables. Comput. Statist. 19 347–366.
- Geiger, D., Meek, C. and Sturmfels, B. (2006). On the toric algebra of graphical models. Ann. Statist. 34 1463–1492.Mathematical Reviews (MathSciNet): MR2278364
Zentralblatt MATH: 1104.60007
Digital Object Identifier: doi:10.1214/009053606000000263
Project Euclid: euclid.aos/1152540755 - Hara, H. and Takemura, A. (2006). Boundary cliques, clique trees and perfect sequences of maximal cliques of a chordal graph. Available at arXiv:cs.DM/0607055.Mathematical Reviews (MathSciNet): MR2266415
- Hara, H. and Takemura, A. (2009). Connecting tables with zero–one entries by a subset of a Markov basis. Available at arXiv:0908.4461.Mathematical Reviews (MathSciNet): MR2485127
Zentralblatt MATH: 1153.62314
Digital Object Identifier: doi:10.1016/j.jspi.2008.07.013 - Hara, H., Takemura, A. and Yoshida, R. (2009). Markov bases for two-way subtable sum problems. J. Pure Appl. Algebra 213 1507–1521.Mathematical Reviews (MathSciNet): MR2517988
Zentralblatt MATH: 05552998
Digital Object Identifier: doi:10.1016/j.jpaa.2008.11.019 - Hoşten, S. and Sullivant, S. (2002). Gröbner bases and polyhedral geometry of reducible and cyclic models. J. Combin. Theory Ser. A 100 277–301.Mathematical Reviews (MathSciNet): MR1940337
Digital Object Identifier: doi:10.1006/jcta.2002.3301
Zentralblatt MATH: 1044.62065 - Irving, R.W. and Jerrum, M.R. (1994). Three-dimensional statistical data security problems. SIAM J. Comput. 23 170–184.Mathematical Reviews (MathSciNet): MR1259001
Zentralblatt MATH: 0802.68046
Digital Object Identifier: doi:10.1137/S0097539790191010 - Lauritzen, S.L. (1996). Graphical Models. Oxford: Oxford Univ. Press.
- Rapallo, F. (2006). Markov bases and structural zeros. J. Symbolic Comput. 41 164–172.Mathematical Reviews (MathSciNet): MR2197152
Zentralblatt MATH: 1120.62044
Digital Object Identifier: doi:10.1016/j.jsc.2005.04.002 - Shibata, Y. (1988). On the tree representation of chordal graphs. J. Graph Theory 12 421–428.Mathematical Reviews (MathSciNet): MR956202
Zentralblatt MATH: 0654.05022
Digital Object Identifier: doi:10.1002/jgt.3190120313 - Sturmfels, B. (1996). Gröbner Bases and Convex Polytopes. Providence, RI: Amer. Math. Soc.Mathematical Reviews (MathSciNet): MR1363949
- Takemura, A. and Aoki, S. (2004). Some characterizations of minimal Markov basis for sampling from discrete conditional distributions. Ann. Inst. Statist. Math. 56 1–17.Mathematical Reviews (MathSciNet): MR2053726
Digital Object Identifier: doi:10.1007/BF02530522
Zentralblatt MATH: 1049.62068 - Takemura, A. and Aoki, S. (2005). Distance reducing Markov bases for sampling from a discrete sample space. Bernoulli 11 793–813.Mathematical Reviews (MathSciNet): MR2172841
Digital Object Identifier: doi:10.3150/bj/1130077594
Project Euclid: euclid.bj/1130077594
Zentralblatt MATH: 1085.62076 - Takemura, A. and Hara, H. (2007). Conditions for swappability of records in a microdata set when some marginals are fixed. Comput. Statist. 22 173–185.Mathematical Reviews (MathSciNet): MR2299254
Zentralblatt MATH: 1199.60033
Digital Object Identifier: doi:10.1007/s00180-007-0024-5 - Vlach, M. (1986). Conditions for the existence of solutions of the three-dimensional planar transportation problem. Discrete Appl. Math. 13 61–78.Mathematical Reviews (MathSciNet): MR829339
Digital Object Identifier: doi:10.1016/0166-218X(86)90069-7
Zentralblatt MATH: 0601.90105 - Zhu, W., Ennis, C.D. and Chen, A. (1998). Many-faceted Rasch modeling expert judgment in test development. Measurement in Physical Education and Exercise Science 2 21–39.
- You have access to this content.
- You have partial access to this content.
- You do not have access to this content.
More like this
- Markov bases for decomposable graphical models
Dobra, Adrian, Bernoulli, 2003 - Positive margins and primary decomposition
Kahle, Thomas, Rauh, Johannes, and Sullivant, Seth, Journal of Commutative Algebra, 2014 - Markov Fields and Log-Linear Interaction Models for Contingency Tables
Darroch, J. N., Lauritzen, S. L., and Speed, T. P., The Annals of Statistics, 1980
- Markov bases for decomposable graphical models
Dobra, Adrian, Bernoulli, 2003 - Positive margins and primary decomposition
Kahle, Thomas, Rauh, Johannes, and Sullivant, Seth, Journal of Commutative Algebra, 2014 - Markov Fields and Log-Linear Interaction Models for Contingency Tables
Darroch, J. N., Lauritzen, S. L., and Speed, T. P., The Annals of Statistics, 1980 - Graphs with Asymptotically Invariant Degree Sequences under Restriction
Cooper, Joshua and Lu, Linyuan, Internet Mathematics, 2011 - A conjugate prior for discrete hierarchical log-linear models
Massam, Hélène, Liu, Jinnan, and Dobra, Adrian, The Annals of Statistics, 2009 - Bayesian learning of weakly structural Markov graph laws using sequential Monte Carlo methods
Olsson, Jimmy, Pavlenko, Tatjana, and Rios, Felix L., Electronic Journal of Statistics, 2019 - A LAYER-DECOMPOSITION WITH DEGREE METHOD FOR SUBGRAPHS RECOGNITION PROBLEM
Xu, Yifan, Taiwanese Journal of Mathematics, 2003 - The maximum likelihood threshold of a path diagram
Drton, Mathias, Fox, Christopher, Käufl, Andreas, and Pouliot, Guillaume, The Annals of Statistics, 2019 - Bayes factors and the geometry of discrete hierarchical loglinear models
Letac, Gérard and Massam, Hélène, The Annals of Statistics, 2012 - Wishart distributions for decomposable graphs
Letac, Gérard and Massam, Hélène, The Annals of Statistics, 2007