Algebraic Statistics and Contingency Table Problems: Log-Linear Models, Likelihood Estimation, and Disclosure Limitation

  • Adrian DobraEmail author
  • Stephen E. Fienberg
  • Alessandro Rinaldo
  • Aleksandra Slavkovic
  • Yi Zhou
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 149)


Contingency tables have provided a fertile ground for the growth of algebraic statistics. In this paper we briefly outline some features of this work and point to open research problems. We focus on the problem of maximum likelihood estimation for log-linear models and a related problem of disclosure limitation to protect the confidentiality of individual responses. Risk of disclosure has often been measured either formally or informally in terms of information contained in marginal tables linked to a log-linear model and has focused on the disclosure potential of small cell counts, especially those equal to 1 or 2. One way to assess the risk is to compute bounds for cell entries given a set of released marginals. Both of these methodologies become complicated for large sparse tables. This paper revisits the problem of computing bounds for cell entries and picks up on a theme first suggested in Fienberg [21] that there is an intimate link between the ideas on bounds and the existence of maximum likelihood estimates, and shows how these ideas can be made rigorous through the underlying mathematics of the same geometric/algebraic framework. We illustrate the linkages through a series of examples. We also discuss the more complex problem of releasing marginal and conditional information. We illustrate the statistical features of the methodology on two examples and then conclude with a series of open problems.

Key words

Conditional tables marginal tables Markov bases maximum likelihood estimate sharp bounds for cell entries toric ideals 


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Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Adrian Dobra
    • 1
    Email author
  • Stephen E. Fienberg
    • 2
  • Alessandro Rinaldo
    • 3
  • Aleksandra Slavkovic
    • 4
  • Yi Zhou
    • 5
  1. 1.Department of StatisticsUniversity of WashingtonSeattleUSA
  2. 2.Department of Statistics, Machine Learning Department and CylabCarnegie Mellon UniversityPittsburghUSA
  3. 3.Department of StatisticsCarnegie Mellon UniversityPittsburghUSA
  4. 4.Department of StatisticsPennsylvania State UniversityUniversity ParkUSA
  5. 5.Machine Learning DepartmentCarnegie Mellon UniversityPittsburghUSA

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