Per citar aquest document: http://ddd.uab.cat/record/18756
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
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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