APPROXIMATING FACES OF MARGINAL POLYTOPES IN DISCRETE HIERARCHICAL MODELS

• 成果类型: Article
• 署名作者: Wang, Nanwei; Rauh, Johannes; Massam, Helene
• 署名单位: York University - Canada; York University - Canada; Max Planck Society
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/18-AOS1710
• 发表日期: 2019
• 页码: 1203-1233
• 关键词: maximum-likelihood-estimation; selection

摘要

The existence of the maximum likelihood estimate in a hierarchical log-linear 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.