Approximating faces of marginal polytopes in discrete hierarchical models

The existence of the maximum likelihood estimate in hierarchical loglinear models 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 sub-models 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 $F_2setminus F_1$. Using both real-world and simulated data, we illustrate our results, and show that our methodology scales to high dimensions.