F-Geometry and Amari’s α-Geometry on a Statistical Manifold
Open Access
- 6 May 2014
- Vol. 16 (5), 2472-2487
- https://doi.org/10.3390/e16052472
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.This publication has 8 references indexed in Scilit:
- Geometry of deformed exponential families: Invariant, dually-flat and conformal geometriesPhysica A: Statistical Mechanics and its Applications, 2012
- Divergence Function, Duality, and Convex AnalysisNeural Computation, 2004
- A Characterization of Monotone and Regular DivergencesAnnals of the Institute of Statistical Mathematics, 1998
- The Role of Differential Geometry in Statistical TheoryInternational Statistical Review, 1986
- Differential geometry of edgeworth expansions in curved exponential familyAnnals of the Institute of Statistical Mathematics, 1983
- Differential Geometry of Curved Exponential Families-Curvatures and Information LossThe Annals of Statistics, 1982
- The Geometry of Exponential FamiliesThe Annals of Statistics, 1978
- Defining the Curvature of a Statistical Problem (with Applications to Second Order Efficiency)The Annals of Statistics, 1975