Dan Geiger
Decision Theory & Adaptive Systems
Microsoft Research
One Microsoft Way
Redmond, Washington, 98052
E-mail: dgeiger@microsoft.com
Phone: 425-703-7389
Fax: 425-936-7329
(On sabbatical from Technion, Computer Science Department, Israel)
David Heckerman
Decision Theory & Adaptive Systems
Microsoft Research
One Microsoft Way
Redmond, Washington, 98052
E-mail: heckerma@microsoft.com
Phone: 425-936-2662
Fax: 425-936-7329
Henry King
Mathematics Department
University of Maryland
College Park, Maryland 20742
E-mail: hck@math.umd.edu
Phone: 301-405-5165
Fax: 301-314-0827
Christopher Meek
Decision Theory & Adaptive Systems
Microsoft Research
One Microsoft Way
Redmond, Washington, 98052
E-mail: meek@microsoft.com
Phone: 425-703-7836
Fax: 425-936-7329
We prove that many graphical models with hidden variables are not curved exponential families. This result, together with the fact that some graphical models are curved and not linear, implies that the hierarchy of graphical models, as linear, curved, and stratified, is non-collapsing; each level in the hierarchy is strictly contained in the larger levels. This result is discussed in the context of model selection of graphical models.
Bayesian networks, Curved exponential families, Manifolds, Model selection, Stratified exponential families
D. Geiger, D. Heckerman, H. King, C. Meek, Stratified Exponential Families: Graphical Models and Model Selection, Technical Report MSR-TR-98-31, Microsoft Research, July, 1998. Submitted to Annals of Statitics. Postscript
D. Geiger, and C. Meek, Graphical Models and Exponential Families, Technical Report MSR-TR-98-10, Microsoft Research, February, 1998. Also in proceedings of UAI-98. Postscript
D. Geiger, D. Heckerman, and C. Meek, Asymptotic Model Selection for Directed Networks with Hidden Variables, Technical Report MSR-TR-96-07, Microsoft Research, May, 1996. Also in proceedings of UAI-96. Postscript