**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.

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