Approximating faces of marginal polytopes in discrete hierarchical models

Abstract

The existence of the maximum likelihood estimate in a hierarchical loglinear model is crucial to the reliability of inference for this model. Determining whether the estimate exists is equivalent to finding whether the sufficient statistics vector t belongs to the boundary of the marginal polytope of the model. The dimension of the smallest face F t containing t determines the dimension of the reduced model which should be considered for correct inference. For higher-dimensional problems, it is not possible to compute F t exactly. Massam and Wang (2015) found an outer approximation to F t using a collection of submodels of the original model. This paper refines the methodology to find an outer approximation and devises a new methodology to find an inner approximation. The inner approximation is given not in terms of a face of the marginal polytope, but in terms of a subset of the vertices of F t . Knowing F t exactly indicates which cell probabilities have maximum likelihood estimates equal to 0. When F t cannot be obtained exactly, we can use, first, the outer approximation F 2 to reduce the dimension of the problem and then the inner approximation F 1 to obtain correct estimates of cell probabilities corresponding to elements of F 1 and improve the estimates of the remaining probabilities corresponding to elements in F 2 \ F 1 . Using both real-world and simulated data, we illustrate our results, and show that our methodology scales to high dimensions.