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

Henryk Gzyl; Lázaro Recht

Publicacions Matemàtiques (2007)

  • Volume: 51, Issue: 2, page 309-332
  • ISSN: 0214-1493

Abstract

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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).

How to cite

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Gzyl, Henryk, and Recht, Lázaro. "Intrinsic geometric on the class of probability densities and exponential families.." Publicacions Matemàtiques 51.2 (2007): 309-332. <http://eudml.org/doc/41897>.

@article{Gzyl2007,
abstract = {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). },
author = {Gzyl, Henryk, Recht, Lázaro},
journal = {Publicacions Matemàtiques},
keywords = {Familia exponencial; Función densidad de probabilidad; Geometría proyectiva; Geometría diferencial global; C*-álgebras; exponential families; geodesic surfaces; parallel transport; Riemannian geometries},
language = {eng},
number = {2},
pages = {309-332},
title = {Intrinsic geometric on the class of probability densities and exponential families.},
url = {http://eudml.org/doc/41897},
volume = {51},
year = {2007},
}

TY - JOUR
AU - Gzyl, Henryk
AU - Recht, Lázaro
TI - Intrinsic geometric on the class of probability densities and exponential families.
JO - Publicacions Matemàtiques
PY - 2007
VL - 51
IS - 2
SP - 309
EP - 332
AB - 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).
LA - eng
KW - Familia exponencial; Función densidad de probabilidad; Geometría proyectiva; Geometría diferencial global; C*-álgebras; exponential families; geodesic surfaces; parallel transport; Riemannian geometries
UR - http://eudml.org/doc/41897
ER -

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