June 2019 Approximating faces of marginal polytopes in discrete hierarchical models
Nanwei Wang, Johannes Rauh, Hélène Massam
Ann. Statist. 47(3): 1203-1233 (June 2019). DOI: 10.1214/18-AOS1710

## Abstract

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 $\mathbf{F}_{t}$ 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 $\mathbf{F}_{t}$ exactly. Massam and Wang (2015) found an outer approximation to $\mathbf{F}_{t}$ 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 $\mathbf{F}_{t}$.

Knowing $\mathbf{F}_{t}$ exactly indicates which cell probabilities have maximum likelihood estimates equal to $0$. When $\mathbf{F}_{t}$ cannot be obtained exactly, we can use, first, the outer approximation $\mathbf{F}_{2}$ to reduce the dimension of the problem and then the inner approximation $\mathbf{F}_{1}$ to obtain correct estimates of cell probabilities corresponding to elements of $\mathbf{F}_{1}$ and improve the estimates of the remaining probabilities corresponding to elements in $\mathbf{F}_{2}\setminus\mathbf{F}_{1}$. Using both real-world and simulated data, we illustrate our results, and show that our methodology scales to high dimensions.

## Citation

Nanwei Wang. Johannes Rauh. Hélène Massam. "Approximating faces of marginal polytopes in discrete hierarchical models." Ann. Statist. 47 (3) 1203 - 1233, June 2019. https://doi.org/10.1214/18-AOS1710

## Information

Received: 1 March 2016; Revised: 1 April 2018; Published: June 2019
First available in Project Euclid: 13 February 2019

zbMATH: 07053506
MathSciNet: MR3911110
Digital Object Identifier: 10.1214/18-AOS1710

Subjects:
Primary: 62F10 , 62F99 , 62H12

Keywords: Existence of the maximum likelihood estimate , extended maximum likelihood estimate , faces , facial sets , marginal polytope