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Entropy 2015, 17(4), 1814-1849; doi:10.3390/e17041814

Geometry of Fisher Information Metric and the Barycenter Map

1
Institute of Mathematics, University of Tsukuba, 1-1-1, Ten-noudai, Tsukuba, 305-8571, Japan
2
Nippon Institute of Technology, Saitama, 345-8501, Japan
This paper is an extended version of our paper published in proceedings, 34th International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering, 21–26 September 2014, Château Clos Lucé, Amboise, France.
*
Author to whom correspondence should be addressed.
Received: 28 January 2015 / Revised: 13 March 2015 / Accepted: 13 March 2015 / Published: 30 March 2015
(This article belongs to the Special Issue Information, Entropy and Their Geometric Structures)
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Abstract

Geometry of Fisher metric and geodesics on a space of probability measures defined on a compact manifold is discussed and is applied to geometry of a barycenter map associated with Busemann function on an Hadamard manifold \(X\). We obtain an explicit formula of geodesic and then several theorems on geodesics, one of which asserts that any two probability measures can be joined by a unique geodesic. Using Fisher metric and thus obtained properties of geodesics, a fibre space structure of barycenter map and geodesical properties of each fibre are discussed. Moreover, an isometry problem on an Hadamard manifold \(X\) and its ideal boundary \(\partial X\)—for a given homeomorphism \(\Phi\) of \(\partial X\) find an isometry of \(X\) whose \(\partial X\)-extension coincides with \(\Phi\)—is investigated in terms of the barycenter map. View Full-Text
Keywords: Fisher metric; probability measure; geodesic; Busemann function; barycenter Fisher metric; probability measure; geodesic; Busemann function; barycenter
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Itoh, M.; Satoh, H. Geometry of Fisher Information Metric and the Barycenter Map. Entropy 2015, 17, 1814-1849.

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