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

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 -