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Computational Information Geometry in Statistics: Theory and Practice
Entropy 2014, 16(5), 2472-2487; doi:10.3390/e16052472

F-Geometry and Amari’s α-Geometry on a Statistical Manifold

*  and *
Indian Institute of Space Science and Technology, Department of Space, Government of India, Valiamala P.O, Thiruvananthapuram-695547, Kerala, India
* Authors to whom correspondence should be addressed.
Received: 13 December 2013 / Revised: 21 April 2014 / Accepted: 25 April 2014 / Published: 6 May 2014
(This article belongs to the Special Issue Information Geometry)
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In this paper, we introduce a geometry called F-geometry on a statistical manifold S using an embedding F of S into the space RX of random variables. Amari’s α-geometry is a special case of F-geometry. Then using the embedding F and a positive smooth function G, we introduce (F,G)-metric and (F,G)-connections that enable one to consider weighted Fisher information metric and weighted connections. The necessary and sufficient condition for two (F,G)-connections to be dual with respect to the (F,G)-metric is obtained. Then we show that Amari’s 0-connection is the only self dual F-connection with respect to the Fisher information metric. Invariance properties of the geometric structures are discussed, which proved that Amari’s α-connections are the only F-connections that are invariant under smooth one-to-one transformations of the random variables.
Keywords: embedding; Amari’s α-connections; F-metric; F-connections; (F,G)-metric; (F,G)-connections; invariance embedding; Amari’s α-connections; F-metric; F-connections; (F,G)-metric; (F,G)-connections; invariance
This is an open access article distributed under the Creative Commons Attribution License (CC BY) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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V., H.K.; S., S.M.K. F-Geometry and Amari’s α-Geometry on a Statistical Manifold. Entropy 2014, 16, 2472-2487.

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