This chapter introduces the fundamental notion of a Markov basis, which represents one of the first connections between commutative algebra and statistics. This connection was made in the paper by Diaconis and the second author  on contingency table analysis. Statistical hypotheses about contingency tables can be tested in an exact approach by performing random walks on a constrained set of tables with non-negative integer entries. Markov bases are of key importance to this statistical methodology because they comprise moves between tables that ensure that the random walk connects every pair of tables in the considered set.
KeywordsContingency Table Hierarchical Model Simplicial Complex Independence Model Integer Lattice
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