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December, 2006 A geometry on the space of probabilities II. Projective spaces and exponential families
Henryk Gzyl , Lázaro Recht
Rev. Mat. Iberoamericana 22(3): 833-849 (December, 2006).

Abstract

In this note we continue a theme taken up in part I, see [Gzyl and Recht: The geometry on the class of probabilities I: The finite dimensional case. Rev. Mat. Iberoamericana 22 (2006), 545-558.], namely to provide a geometric interpretation of exponential families as end points of geodesics of a non-metric connection in a function space. For that we characterize the space of probability densities as a projective space in the class of strictly positive functions, and these will be regarded as a homogeneous reductive space in the class of all bounded complex valued functions. We shall develop everything in a generic $\mathcal{C}^*$-algebra setting, but shall have the function space model in mind.

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Henryk Gzyl . Lázaro Recht . "A geometry on the space of probabilities II. Projective spaces and exponential families." Rev. Mat. Iberoamericana 22 (3) 833 - 849, December, 2006.

Information

Published: December, 2006
First available in Project Euclid: 22 January 2007

zbMATH: 1122.46039
MathSciNet: MR2320403

Subjects:
Primary: 46L05 , 53C05 , 53C56 , 60B99 , 60E05
Secondary: 32M99 , 53C30 , 62A25 , 94A17

Keywords: $C^*$-algebra , exponential families , lifting of geodesics , maximum entropy method , reductive homogeneous space

Rights: Copyright © 2006 Departamento de Matemáticas, Universidad Autónoma de Madrid

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