Intrinsic geometric on the class of probability densities and exponential families.

Título inglés	Intrinsic geometric on the class of probability densities and exponential families.
Título español	Geometría intrínseca en la clase de densidades de probabilidad y familias exponenciales.
Autor/es	Gzyl, Henryk ; Recht, Lázaro
Organización	Dep. Cómput. Cien. Estadíst. Univ. Simón Bolívar, Caracas, Venezuela;Dep. Mat. Univ. Simón Bolívar, Caracas, Venezuela
Revista	0214-1493
Publicación	2007, 51 (2): 309-332, 23 Ref.
Tipo de documento	articulo
Idioma	Inglés
Resumen inglés	We present a way of thinking of exponential farnilies 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 rneasure 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).
Clasificación UNESCO	120404 ; 120907
Palabras clave español	Familia exponencial ; Función densidad de probabilidad ; Geometría proyectiva ; Geometría diferencial global ; C*-álgebras
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