Department of Statistics Unitmark
Dietrich College of Humanities and Social Sciences

On Maximum Likelihood Estimation in Log-Linear Models

Publication Date

May, 2005

Publication Type

Tech Report


Alessandro Rinaldo


In this article, we combine results from the theory of linear exponential families, polyhedral geometry and algebraic geometry to provide analytic and geometric characterizations of log-linear models and maximum likelihood estimation. Geometric and combinatorial conditions for the existence of the Maximum Likelihood Estimate (MLE) of the cell mean vector of a contingency table are given for general log-linear models under conditional Poisson sampling. It is shown that any log-linear model can be generalized to an extended exponential family of distributions parametrized, in a mean value sense, by points of a polyhedron. Such a parametrization is continuous and, with respect to this extended family, the MLE always exists and is unique. In addition, the set of cell mean vectors form a subset of a toric variety consisting of non-negative points satisfying a certain system of polynomial equations. These results of are theoretical and practical importance for estimation and model selection.