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