Intrinsic geometry on the class of probability densities and exponential families
Gzyl, Henryk (Centro de Finanzas Iesa (Caracas, Venezuela))
Recht, Lázaro (Universidad Simón Bolívar (Veneçuela). 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).
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Llengua:	Anglès
Document:	Article ; recerca ; Versió publicada
Matèria:	Exponential families ; Projective geometry ; Parallel transport ; Sequences of convolution type
Publicat a:	Publicacions matemàtiques, V. 51 n. 2 (2007) p. 309-322, ISSN 2014-4350

Adreça alternativa: https://raco.cat/index.php/PublicacionsMatematiques/article/view/218491
DOI: 10.5565/PUBLMAT_51207_03

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