Revista Matemática Iberoamericana

A geometry on the space of probabilities II. Projective spaces and exponential families

Henryk Gzyl and Lázaro Recht

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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 $\mathcal{C}^*$-algebra setting, but shall have the function space model in mind.

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Rev. Mat. Iberoamericana, Volume 22, Number 3 (2006), 833-849.

First available in Project Euclid: 22 January 2007

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Primary: 46L05: General theory of $C^*$-algebras 53C05: Connections, general theory 53C56: Other complex differential geometry [See also 32Cxx] 60B99: None of the above, but in this section 60E05: Distributions: general theory
Secondary: 53C30: Homogeneous manifolds [See also 14M15, 14M17, 32M10, 57T15] 32M99: None of the above, but in this section 62A25 94A17: Measures of information, entropy

$C^*$-algebra reductive homogeneous space lifting of geodesics exponential families maximum entropy method


Gzyl, Henryk; Recht, Lázaro. A geometry on the space of probabilities II. Projective spaces and exponential families. Rev. Mat. Iberoamericana 22 (2006), no. 3, 833--849.

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