F-Geometry and Amari’s α-Geometry on a Statistical Manifold
AbstractIn 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.
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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.
V. HK, S. SMK. F-Geometry and Amari’s α-Geometry on a Statistical Manifold. Entropy. 2014; 16(5):2472-2487.Chicago/Turabian Style
V., Harsha K.; S., Subrahamanian M.K. 2014. "F-Geometry and Amari’s α-Geometry on a Statistical Manifold." Entropy 16, no. 5: 2472-2487.