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Intrinsic geometry on the class of probability densities and exponential families

Henryk Gzyl and Lázaro Recht

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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 $\ell_1(\alpha)$.

Article information

Publ. Mat., Volume 51, Number 2 (2007), 309-332.

First available in Project Euclid: 31 July 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46L05: General theory of $C^*$-algebras 53C05: Connections, general theory 53C56: Other complex differential geometry [See also 32Cxx] 62B01 60B99: None of the above, but in this section 60E05: Distributions: general theory
Secondary: 53C30: Homogeneous manifolds [See also 14M15, 14M17, 32M10, 57T15] 51M05: Euclidean geometries (general) and generalizations 55M 62A25 22E 33E

Exponential families projective geometry parallel transport sequences of convolution type


Gzyl, Henryk; Recht, Lázaro. Intrinsic geometry on the class of probability densities and exponential families. Publ. Mat. 51 (2007), no. 2, 309--332.

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