On Maximum Likelihood Estimation in Log-Linear Models

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.