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

In this paper, we introduce a geometry called F-geometry on a statistical manifold S using an embedding F of S into the space R-X of random variables. Amari's alpha-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 alpha-connections are the only F connections that are invariant under smooth one-to-one transformations of the random variables.