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 |
Acceso al artículo completo |