Open Access
2009 On the geometry of discrete exponential families with application to exponential random graph models
Alessandro Rinaldo, Stephen E. Fienberg, Yi Zhou
Electron. J. Statist. 3: 446-484 (2009). DOI: 10.1214/08-EJS350


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.


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Alessandro Rinaldo. Stephen E. Fienberg. Yi Zhou. "On the geometry of discrete exponential families with application to exponential random graph models." Electron. J. Statist. 3 446 - 484, 2009.


Published: 2009
First available in Project Euclid: 26 May 2009

zbMATH: 1326.62071
MathSciNet: MR2507456
Digital Object Identifier: 10.1214/08-EJS350

Primary: 62F99
Secondary: 62F99

Keywords: exponential families , exponential random graphs , maximum likelihood estimates , normal cone

Rights: Copyright © 2009 The Institute of Mathematical Statistics and the Bernoulli Society

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