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 -algebra setting, but shall have the function space model in mind.
Cite this article
Henryk Gzyl, Lázaro Recht, A geometry on the space of probabilities II. Projective spaces and exponential families. Rev. Mat. Iberoam. 22 (2006), no. 3, pp. 833–849