Polyhedral conditions for the nonexistence of the MLE for hierarchical log-linear models

Authors:
Nicholas Eriksson;Stephen E. Fienberg;Alessandro Rinaldo;Seth Sullivant
Affiliations:
Department of Mathematics, University of California, Berkeley, United States;Department of Statistics, Carnegie Mellon University, United States and Center for Automated Learning and Discovery and Cylab, Carnegie Mellon University, United States;Department of Statistics, Carnegie Mellon University, United States;Department of Mathematics, University of California, Berkeley, United States
Venue:
Journal of Symbolic Computation
Year:
2006

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Abstract

We provide a polyhedral description of the conditions for the existence of the maximum likelihood estimate (MLE) for a hierarchical log-linear model. The MLE exists if and only if the observed margins lie in the relative interior of the marginal cone. Using this description, we give an algorithm for determining if the MLE exists. If the tree width is bounded, the algorithm runs in polynomial time. We also perform a computational study of the case of three random variables under the no three-factor effect model.