On the geometry of discrete exponential families with application to exponential random graph models
On the geometry of discrete exponential families with application to exponential random graph models
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DOI：
10.1214/08-ejs350
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发表时间：
2009-01-01
影响因子：
1.1
通讯作者：
Zhou, Yi
中科院分区：
文献类型：
--
作者：
Rinaldo, Alessandro;Fienberg, Stephen E.;Zhou, Yi
人们对用于分析网络数据的统计模型的兴趣激增，对指数随机图 (ERG) 模型类也产生了极大的兴趣，特别是在计算最大似然估计的困难方面。与这些困难相关的问题与更广泛的离散指数族结构有关。本文分两部分重新审视这些问题。首先，我们考虑具有离散基测度和多面体凸支持 P 的 k 维指数分布族的闭包。我们证明 P 的法向扇形是一个几何对象，在推导相应扩展指数族的统计和几何特性方面发挥着基础作用。我们从理论和计算的角度讨论了它与最大似然估计的相关性。其次，我们将我们的结果应用于 ERG 模型的分析。通过详细的例子，我们提供了 ERG 模型属性的一些表征，特别是 ERG 模型的某些行为（称为简并性）。
There has been an explosion of interest in statistical models for analyzing network data, and considerable interest in the class of exponential random graph (ERG) models, especially in connection with difficulties in computing maximum likelihood estimates. The issues associated with these difficulties relate to the broader structure of discrete exponential families. This paper re-examines the issues in two parts. First we consider the closure of k-dimensional exponential families of distribution with discrete base measure and polyhedral convex support P. We show that the normal fan of P is a geometric object that plays a fundamental role in deriving the statistical and geometric properties of the corresponding extended exponential families. We discuss its relevance to maximum likelihood estimation, both from a theoretical and computational standpoint. Second, we apply our results to the analysis of ERG models. By means of a detailed example, we provide some characterization of the properties of ERG models, and, in particular, of certain behaviors of ERG models known as degeneracy.