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).
@article{urn:eudml:doc:41897,
title = {Intrinsic geometric on the class of probability densities and exponential families.},
journal = {Publicacions Matem\`atiques},
year = {2007},
pages = {309-332},
zbl = {1141.46025},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:41897}
}
Gzyl, Henryk; Recht, Lázaro. Intrinsic geometric on the class of probability densities and exponential families.. Publicacions Matemàtiques, (2007), pp. 309-332. http://gdmltest.u-ga.fr/item/urn:eudml:doc:41897/