**Nick Eriksson, Stephen E. Fienberg, Alessandro Rinaldo and
Seth Sullivant**

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.

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