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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
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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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