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April 2001 Stratified exponential families: Graphical models and model selection
Dan Geiger, David Heckerman, Henry King, Christopher Meek
Ann. Statist. 29(2): 505-529 (April 2001). DOI: 10.1214/aos/1009210550


We describe a hierarchy of exponential families which is useful for distinguishing types of graphical models. Undirected graphical models with no hidden variables are linear exponential families (LEFs). Directed acyclic graphical (DAG) models and chain graphs with no hidden variables, includ­ ing DAG models with several families of local distributions, are curved exponential families (CEFs). Graphical models with hidden variables are what we term stratified exponential families (SEFs). A SEF is a finite union of CEFs of various dimensions satisfying some regularity conditions. We also show that this hierarchy of exponential families is noncollapsing with respect to graphical models by providing a graphical model which is a CEF but not a LEF and a graphical model that is a SEF but not a CEF. Finally, we show how to compute the dimension of a stratified exponential family. These results are discussed in the context of model selection of graphical models.


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Dan Geiger. David Heckerman. Henry King. Christopher Meek. "Stratified exponential families: Graphical models and model selection." Ann. Statist. 29 (2) 505 - 529, April 2001.


Published: April 2001
First available in Project Euclid: 24 December 2001

zbMATH: 1012.62012
MathSciNet: MR1863967
Digital Object Identifier: 10.1214/aos/1009210550

Primary: 60E05, 62H05

Rights: Copyright © 2001 Institute of Mathematical Statistics


Vol.29 • No. 2 • April 2001
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