On the Existence of the MLE for Log-Linear Models

Log-Linear models are a powerful statistical tool for the analysis of categorical data. Their use has increased greatly over the past two decades with the compilation and distribution of large, sparse databases in the social and medical sciences as well as in machine learning applications. Such databases typically take the form of high-dimensional contingency tables with a disproportionately large number of empty cells and small counts. In these cases, the excessive degree of sparsity of the data may compromise the feasibility and correctness of log-linear methodologies, as well as of most off-the-shelf statistical procedures. In particular, the Maximum Likelihood Estimate (MLE) of the cell mean vector, whose existence is crucial for assessment of fit, model selection and interpretation, is very likely to be undefined. This talk is concerned with the problem of non-existence of the MLE caused by the presence of sampling zeros in the table. Available results in the statistical literature are non-constructive and impractical, as they do not lead directly to implementable numerical procedures, nor do they suggest alternative methods of inference. Following the recent advances in the field of algebraic statistics, I will illustrate a more general approach to the study of log-linear models that takes advantage of the connections between algebraic and polyhedral geometry and the theory of exponential families. I will describe novel geometric and combinatorial conditions for the existence of the MLE that generalize existing results and are also of practical use. I will introduce the notion of Extended MLE and illustrate its meaning and use with various examples.