In this note we continue a theme taken up in part I, see [Gzyl
and Recht: The geometry on the class of probabilities I: The
finite dimensional case. Rev. Mat. Iberoamericana 22 (2006),
545-558.], namely to provide a geometric interpretation of
exponential families as end points of geodesics of a non-metric
connection in a function space. For that we characterize the
space of probability densities as a projective space in the class
of strictly positive functions, and these will be regarded as a
homogeneous reductive space in the class of all bounded complex
valued functions. We shall develop everything in a generic
$\mathcal{C}^*$-algebra setting, but shall have the function
space model in mind.
Publié le : 2006-12-14
Classification:
$C^*$-algebra,
reductive homogeneous space,
lifting of geodesics,
exponential families,
maximum entropy method,
46L05,
53C05,
53C56,
60B99,
60E05,
53C30,
32M99,
62A25,
94A17
@article{1169480032,
author = {Gzyl
,
Henryk and Recht
,
L\'azaro},
title = {A geometry on the space of probabilities II. Projective spaces and exponential families},
journal = {Rev. Mat. Iberoamericana},
volume = {22},
number = {2},
year = {2006},
pages = { 833-849},
language = {en},
url = {http://dml.mathdoc.fr/item/1169480032}
}
Gzyl
,
Henryk; Recht
,
Lázaro. A geometry on the space of probabilities II. Projective spaces and exponential families. Rev. Mat. Iberoamericana, Tome 22 (2006) no. 2, pp. 833-849. http://gdmltest.u-ga.fr/item/1169480032/