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

Publicacions Matemàtiques (2007)

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

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topGzyl, 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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