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Intrinsic geometry on the class of probability densities and exponential families
Gzyl, Henryk (Centro de Finanzas Iesa (Caracas, Venezuela))
Recht, Lázaro (Universidad Simón Bolívar (Caracas, Venezuela). Departamento de Matemáticas)

Data: 2007
Resum: We present a way of thinking of exponential families as geodesic surfaces in the class of positive functions considered as a (multiplicative) sub-group G+ of the group G of all invertible elements in the algebra A of all complex bounded functions defined on a measurable space. For that we have to study a natural geometry on that algebra. The class D of densities with respect to a given measure will happen to be representatives of equivalence classes defining a projective space in A. The natural geometry is defined by an intrinsic group action which allows us to think of the class of positive, invertible functions G+ as a homogeneous space. Also, the parallel transport in G+ and D will be given by the original group action. Besides studying some relationships among these constructions, we examine some Riemannian geometries and provide a geometric interpretation of Pinsker’s and other classical inequalities. Also we provide a geometric reinterpretation of some relationships between polynomial sequences of convolution type, probability distributions on N in terms of geodesics in the Banach space ℓ1(α).
Drets: Tots els drets reservats.
Llengua: Anglès.
Document: article ; recerca ; publishedVersion
Matèria: Exponential families ; Projective geometry ; Parallel transport ; Sequences of convolution type
Publicat a: Publicacions Matemàtiques, V. 51 n. 2 (2007) p. 309-322, ISSN 0214-1493

DOI: 10.5565/PUBLMAT_51207_03

24 p, 249.3 KB

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