# Fibers of multi-way contingency tables given conditionals: relation to marginals, cell bounds and Markov bases

Aleksandra Slavković, Xiaotian Zhu, Sonja Petrović

Research output: Contribution to journalArticle

5 Citations (Scopus)

### 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) 621-648 28 Annals of the Institute of Statistical Mathematics 67 4 https://doi.org/10.1007/s10463-014-0471-z Published - Aug 22 2015

### Fingerprint

Markov Basis
Contingency Table
Fiber
Cell
Disclosure
Algebraic Statistics
Linear Diophantine equation
Conditional Inference
Exact Inference
Directed Acyclic Graph
Solution Set
Tables
Table
Sample Size
Decompose

### All Science Journal Classification (ASJC) codes

• Statistics and Probability

### Cite this

title = "Fibers of multi-way contingency tables given conditionals: relation to marginals, cell bounds and Markov bases",
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.",
author = "Aleksandra Slavković and Xiaotian Zhu and Sonja Petrović",
year = "2015",
month = "8",
day = "22",
doi = "10.1007/s10463-014-0471-z",
language = "English (US)",
volume = "67",
pages = "621--648",
journal = "Annals of the Institute of Statistical Mathematics",
issn = "0020-3157",
publisher = "Springer Netherlands",
number = "4",

}

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

In: Annals of the Institute of Statistical Mathematics, Vol. 67, No. 4, 22.08.2015, p. 621-648.

Research output: Contribution to journalArticle

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

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JO - Annals of the Institute of Statistical Mathematics

JF - Annals of the Institute of Statistical Mathematics

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