Abstract
For general (i.e., non decomposable) hierarchical models of contingency tables, structure of a Markov basis is very complicated. It contains moves of higher degree in general and the complexity increases as the sizes of contingency tables increase. However, Markov chain Monte Carlo methods for non-decomposable models are very useful since the closed form expression of null distributions cannot be obtained for general non-decomposable models and therefore other methods such as exact methods cannot be used. In this chapter, we first consider the no-three-factor interaction model of three-way contingency tables as an example of non-decomposable models. The complete structure of Markov bases for this model has not yet been obtained at present. We give a structure of minimal Markov bases for 3 ×3 ×K contingency tables with fixed two-dimensional marginals. We then discuss Markov bases of reducible models, Markov complexity and Markov width.
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Aoki, S., Hara, H., Takemura, A. (2012). Markov Basis for No-Three-Factor Interaction Models and Some Other Hierarchical Models. In: Markov Bases in Algebraic Statistics. Springer Series in Statistics, vol 199. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-3719-2_9
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DOI: https://doi.org/10.1007/978-1-4614-3719-2_9
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