Maximum Likelihood Estimation in Log-Linear Models Supplementary Material

This document contains supplementary material to the article “Maximum Likelihood Estimation in LogLinear Models” by S.E. Fienberg and A. Rinaldo, henceforth referred to as FR. In section 2 we provide the proofs to some of the results announced in the article. Throughout, we assume familiarity with basic notions of polyhedral geometry: see Ziegler (1998), Schrijver (1998) and Rockafellar (1970) for in-depth treatments, and section 2.1 of Rinaldo et al. (2009) for a brief review of the concepts directly relevant to our setting. In section 3 we develop and analyze algorithms for carrying out a number of tasks related to computing extended MLEs, determining facial sets and adjusting the number of degrees of freedom under a nonexistent MLE for Poisson, multinomial and product multinomial likelihoods. Some of these algorithms are implemented in a MATLAB toolbox available at http://www.stat.cmu.edu/~arinaldo/ExtMLE/. Due to space limitations, the algorithmic part of our work could only be described in this supplementary file. Nonetheless, we regard this as an integral component of our analysis on inference in log0linear models, equally important as the theoretical results we provide in the article.