Markov basis and Gröbner basis of Segre–Veronese configuration for testing independence in group-wise selections

Abstract

We consider testing independence in group-wise selections with some restrictions on combinations of choices. We present models for frequency data of selections for which it is easy to perform conditional tests by Markov chain Monte Carlo (MCMC) methods. When the restrictions on the combinations can be described in terms of a Segre-Veronese configuration, an explicit form of a Gröbner basis consisting of binomials of degree two is readily available for performing a Markov chain. We illustrate our setting with the National Center Test for university entrance examinations in Japan. We also apply our method to testing independence hypotheses involving genotypes at more than one locus or haplotypes of alleles on the same chromosome.

A Gröbner basis characterization for chordal comparability graphs

“…Thus, in general, I M d (P ) is a toric ideal of a subconfiguration of the dth Veronese subring. There are several results on toric ideals of subconfigurations of the dth Veronese subring: algebras of Veronese type [20,Theorem 14.2] and algebras of Segre-Veronese type [17,1]. However, the results of the present paper are different from these results.…”

“…See [11,20]. The toric ideals of algebras of Veronese type [20,Theorem 14.2] and algebras of Segre-Veronese type [17,1] have a squarefree initial ideal. On the other hand, I M d (P ) has no squarefree initial ideal except for some trivial cases.…”

A Quick Introduction to Gröbner Bases

“…We define the total order < rev on M n by setting u < rev v if either (i) P n iD1 a i < P n iD1 b i , or (ii) P n iD1 a i D P n iD1 b i and the rightmost nonzero component of the vector .b 1 a 1 ; b 2 a 2 ; : : : ; b n a n / is negative. It follows that < rev is a monomial order on KOEx, which is called the reverse lexicographic order 3 n be monomials. We define the total order < purelex on M n by setting u < purelex v if the leftmost nonzero component of the vector .b 1 a 1 ; b 2 a 2 ; : : : ; b n a n / is positive.…”

“…Algebraic statistics supplies commutative algebra with new problems [2,21]. Conversely, toric ideals studied in commutative algebra supply algebraic statistics with new statistical models [3,20]. The interrelationship between algebraic statistics and commutative algebra is worth studying hardly.…”

Conditional inference given partial information in contingency tables using Markov bases

“…Following the first line of research, Takemura and Aoki42 characterize the minimal Markov bases needed for sampling from discrete conditional distributions, Aoki et al43 explicitly describe Markov bases for constraint matrices whose generating ideals can be described by Segre‐Veronese configurations. Dobra and Sullivant44 provide a strategy for computing Markov bases for reducible log‐linear models, and Ogawa and Takemura45 for block effects models in two‐way tables.…”