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)

Date:	2007
Abstract:	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).
Rights:	Aquest material està protegit per drets d'autor i/o drets afins. Podeu utilitzar aquest material en funció del que permet la legislació de drets d'autor i drets afins d'aplicació al vostre cas. Per a d'altres usos heu d'obtenir permís del(s) titular(s) de drets.
Language:	Anglès
Document:	Article ; recerca ; Versió publicada
Subject:	Exponential families ; Projective geometry ; Parallel transport ; Sequences of convolution type
Published in:	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

24 p, 249.3 KB

The record appears in these collections:
Articles > Published articles > Publicacions matemàtiques
Articles > Research articles