### Abstract

A fiber of a contingency table is the space of all realizations of the table under a given set of constraints such as marginal totals. Understanding the geometry of this space is a key problem in algebraic statistics, important for conducting exact conditional inference, calculating cell bounds, imputing missing cell values, and assessing the risk of disclosure of sensitive information. Motivated by disclosure problems, in this paper we study the space of all possible tables for a given sample size and set of observed conditional frequencies. We show that this space can be decomposed according to different possible marginals, which, in turn, are encoded by the solution set of a linear Diophantine equation. Our decomposition has two important consequences: (1) we derive new cell bounds, some including connections to directed acyclic graphs, and (2) we describe a structure for the Markov bases for the given space that leads to a simplified calculation of Markov bases in this particular setting.

Original language | English (US) |
---|---|

Pages (from-to) | 621-648 |

Number of pages | 28 |

Journal | Annals of the Institute of Statistical Mathematics |

Volume | 67 |

Issue number | 4 |

DOIs | |

State | Published - Aug 22 2015 |

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### All Science Journal Classification (ASJC) codes

- Statistics and Probability

### Cite this

*Annals of the Institute of Statistical Mathematics*,

*67*(4), 621-648. https://doi.org/10.1007/s10463-014-0471-z

}

*Annals of the Institute of Statistical Mathematics*, vol. 67, no. 4, pp. 621-648. https://doi.org/10.1007/s10463-014-0471-z

**Fibers of multi-way contingency tables given conditionals : relation to marginals, cell bounds and Markov bases.** / Slavković, Aleksandra; Zhu, Xiaotian; Petrović, Sonja.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Fibers of multi-way contingency tables given conditionals

T2 - relation to marginals, cell bounds and Markov bases

AU - Slavković, Aleksandra

AU - Zhu, Xiaotian

AU - Petrović, Sonja

PY - 2015/8/22

Y1 - 2015/8/22

N2 - A fiber of a contingency table is the space of all realizations of the table under a given set of constraints such as marginal totals. Understanding the geometry of this space is a key problem in algebraic statistics, important for conducting exact conditional inference, calculating cell bounds, imputing missing cell values, and assessing the risk of disclosure of sensitive information. Motivated by disclosure problems, in this paper we study the space of all possible tables for a given sample size and set of observed conditional frequencies. We show that this space can be decomposed according to different possible marginals, which, in turn, are encoded by the solution set of a linear Diophantine equation. Our decomposition has two important consequences: (1) we derive new cell bounds, some including connections to directed acyclic graphs, and (2) we describe a structure for the Markov bases for the given space that leads to a simplified calculation of Markov bases in this particular setting.

AB - A fiber of a contingency table is the space of all realizations of the table under a given set of constraints such as marginal totals. Understanding the geometry of this space is a key problem in algebraic statistics, important for conducting exact conditional inference, calculating cell bounds, imputing missing cell values, and assessing the risk of disclosure of sensitive information. Motivated by disclosure problems, in this paper we study the space of all possible tables for a given sample size and set of observed conditional frequencies. We show that this space can be decomposed according to different possible marginals, which, in turn, are encoded by the solution set of a linear Diophantine equation. Our decomposition has two important consequences: (1) we derive new cell bounds, some including connections to directed acyclic graphs, and (2) we describe a structure for the Markov bases for the given space that leads to a simplified calculation of Markov bases in this particular setting.

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U2 - 10.1007/s10463-014-0471-z

DO - 10.1007/s10463-014-0471-z

M3 - Article

AN - SCOPUS:84931566233

VL - 67

SP - 621

EP - 648

JO - Annals of the Institute of Statistical Mathematics

JF - Annals of the Institute of Statistical Mathematics

SN - 0020-3157

IS - 4

ER -