A geometry on the space of probabilities. I: The finite dimensional case. (English) Zbl 1121.46043

Gzyl, Henryk; Recht, Lázaro

Rev. Mat. Iberoam. 22, No. 2, 545-558 (2006).

Summary: We provide a natural way of defining exponential coordinates on the class of probabilities on the set \(\Omega = [1,n]\) or on \(\mathbb{P} = \{p=(p_1,\dots,p_n)\in \mathbb{R}^n \mid p_i > 0;\;\Sigma_{i=1}^n p_i =1\}\). For that, we have to regard \(\mathbb{P}\) as a projective space and the exponential coordinates will be related to geodesic flows in \(\mathbb{C}^n\).
[Part II has appeared ibid.22, No.3, 833–849 (2006; Zbl 1122.46039).]

MSC:

46L05	General theory of \(C^*\)-algebras
28A33	Spaces of measures, convergence of measures
46L99	Selfadjoint operator algebras (\(C^*\)-algebras, von Neumann (\(W^*\)-) algebras, etc.)
94A17	Measures of information, entropy

Keywords:

\(C^*\)-algebra; reductive homogeneous space; lifting of geodesics; exponential families; maximum entropy method

Citations:

Zbl 1122.46039

References:

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