Open Access
2007 Intrinsic geometry on the class of probability densities and exponential families
Henryk Gzyl, Lázaro Recht
Publ. Mat. 51(2): 309-332 (2007).


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 $\ell_1(\alpha)$.


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Henryk Gzyl. Lázaro Recht. "Intrinsic geometry on the class of probability densities and exponential families." Publ. Mat. 51 (2) 309 - 332, 2007.


Published: 2007
First available in Project Euclid: 31 July 2007

zbMATH: 1141.46025
MathSciNet: MR2334793

Primary: 46L05 , 53C05 , 53C56 , 60B99 , 60E05 , 62B01
Secondary: 22E , 33E , 51M05 , 53C30 , 55M , 62A25

Keywords: exponential families , parallel transport , projective geometry , sequences of convolution type

Rights: Copyright © 2007 Universitat Autònoma de Barcelona, Departament de Matemàtiques

Vol.51 • No. 2 • 2007
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