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 1121.46043References:
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