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A geometry on the space of probabilities. II: Projective spaces and exponential families. (English) Zbl 1122.46039

Summary: We continue a theme taken up in Part I, see [H. Gzyl and L. Recht, Rev. Mat. Iberom. 22, 545–558 (2006; Zbl 1121.46043)], namely, to provide a geometric interpretation of exponential families as endpoints 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 develop everything in a generic \({\mathcal C}^*\)-algebra setting, but have the function space model in mind.

MSC:

46L05 General theory of \(C^*\)-algebras
53C05 Connections (general theory)
53C56 Other complex differential geometry
60B99 Probability theory on algebraic and topological structures
60E05 Probability distributions: general theory
53C30 Differential geometry of homogeneous manifolds
32M99 Complex spaces with a group of automorphisms
62A01 Foundations and philosophical topics in statistics
94A17 Measures of information, entropy
58B20 Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds
28A33 Spaces of measures, convergence of measures

Citations:

Zbl 1121.46043
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References:

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