Abstract
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.
Highlights
Geometric study of statistical estimation has opened up an interesting new area called the InformationGeometry
We show that F −metric is the Fisher information metric and Amari’s α−geometry is a special case of F −geometry
We prove that the Fisher information metric and Amari’s α−connections are invariant under both the transformation of the parameter and the transformation of the random variable
Summary
Amari [2] defined a one parameter family of functions called the α−embeddings given by ( These Γαijk uniquely determine affine connections ∇α on the statistical manifold S by. Burbea [14] introduced the concept of weighted Fisher information metric using a positive continuous function We use this idea to define weighted F −metric and weighted F −connections. When G(p) = 1, (F, G)−connection reduces to the F −connection and the metric F,G reduces to the Fisher information metric This is a more general way of defining Riemannian metrics and affine connections on a statistical manifold
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