Intrinsic geometry on the class of probability densities and exponential families. (English) Zbl 1141.46025

Gzyl, Henryk; Recht, Lázaro

Publ. Mat., Barc. 51, No. 2, 309-332 (2007).

Summary: We present a way of thinking of exponential families 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 \({\mathcal A}\) of all complex bounded functions defined on a measurable space. For that we have to study a natural geometry on that algebra. The class \({\mathcal D}\) of densities with respect to a given measure will happen to be representatives of equivalence classes defining a projective space in \({\mathcal A}\). 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 \({\mathcal D}\) 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 \(\mathbb N\) in terms of geodesics in the Banach space \(\ell_1(\alpha)\).

MSC:

46L05	General theory of \(C^*\)-algebras
53C05	Connections (general theory)
53C56	Other complex differential geometry

Keywords:

exponential families; geodesic surfaces; parallel transport; Riemannian geometries