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**Distance-reducing Markov bases for sampling from a discrete sample space.**
*(English)*
Zbl 1085.62076

From the introduction: A Markov basis for sampling from a discrete conditional distribution is usually studied in the framework of a toric ideal and its Gröbner basis. For the case of a \(3\times 3\times K\) contingency table with fixed two-dimensional marginals, in [Aust. N. Z. J. Stat. 45, No. 2, 229–249 (2003; Zbl 1064.62068)] we used a more elementary approach to derive a unique minimal Markov basis. The approach was based on exhaustive consideration of sign patterns when the \(L_1\)-norm (1-norm) between two contingency tables with the same margmals is minimized. In order to prove that a candidate set \(\mathcal B\) of moves is a Markov basis, we have shown that the 1-norm between two contingency tables can always be decreased by an element of \(\mathcal B\).

In order to study a minimal Markov basis and its uniqueness for other models, in [Ann. Inst. Stat. Math. 56, No. 1, 1–17 (2004; Zbl 1049.62068)] we considered whether two elements of the same fibre (reference set) are mutually accessible by a set of lower-degree moves and derived some results on the characterization of minimal Markov bases. Note that the notion of mutual accessibility is not directly related to any metric on the fibres. Therefore although the approaches in these two papers were similar, they were different in explicit consideration of the metric on the fibres.

In this paper we explicitly consider the 1-norm on the fibres in the general framework of our paper from 2004 and derive some characterizations of 1-norm-reducing Markov bases.

In order to study a minimal Markov basis and its uniqueness for other models, in [Ann. Inst. Stat. Math. 56, No. 1, 1–17 (2004; Zbl 1049.62068)] we considered whether two elements of the same fibre (reference set) are mutually accessible by a set of lower-degree moves and derived some results on the characterization of minimal Markov bases. Note that the notion of mutual accessibility is not directly related to any metric on the fibres. Therefore although the approaches in these two papers were similar, they were different in explicit consideration of the metric on the fibres.

In this paper we explicitly consider the 1-norm on the fibres in the general framework of our paper from 2004 and derive some characterizations of 1-norm-reducing Markov bases.

### MSC:

62H17 | Contingency tables |

62D05 | Sampling theory, sample surveys |

05C90 | Applications of graph theory |

62B05 | Sufficient statistics and fields |

### Software:

4ti2
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\textit{A. Takemura} and \textit{S. Aoki}, Bernoulli 11, No. 5, 793--813 (2005; Zbl 1085.62076)

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### References:

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