Revista Matemática Iberoamericana


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Volume 22, Issue 3, 2006, pp. 833–849
DOI: 10.4171/RMI/475

Published online: 2006-12-31

A geometry on the space of probabilities II. Projective spaces and exponential families

Henryk Gzyl[1] and Lázaro Recht[2]

(1) Universidad Carlos III, Madrid-Getafe, Spain
(2) Universidad Simón Bolívar, Caracas, Venezuela

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.

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

Gzyl Henryk, Recht Lázaro: A geometry on the space of probabilities II. Projective spaces and exponential families. Rev. Mat. Iberoam. 22 (2006), 833-849. doi: 10.4171/RMI/475