On the closure of relational models

TL;DR

This paper explores the theoretical properties of maximum likelihood estimators in relational models for contingency tables, especially when data contain zeros, establishing conditions for existence and methods for computation.

Contribution

It characterizes the closure of relational models with respect to Bregman divergence and demonstrates the existence of unique MLEs as limits of sequences within the model.

Findings

Unique MLEs always exist as limits in the model closure.

The closure of the model is characterized by Bregman divergence.

Iterative scaling can compute MLEs in both the model and its closure.

Abstract

Relational models for contingency tables are generalizations of log-linear models, allowing effects associated with arbitrary subsets of cells in a possibly incomplete table, and not necessarily containing the overall effect. In this generality, the MLEs under Poisson and multinomial sampling are not always identical. This paper deals with the theory of maximum likelihood estimation in the case when there are observed zeros in the data. A unique MLE to such data is shown to always exist in the set of pointwise limits of sequences of distributions in the original model. This set is equal to the closure of the original model with respect to the Bregman information divergence. The same variant of iterative scaling may be used to compute the MLE in the original model and in its closure.