Infinite Dimensional Analysis Quantum Probability and Related Topics · 1998 · 88 citations · 15 references
Natural SciencesMathematicsGeometryStatisticsTopologyDifferential GeometryFiber BundlesGeometric TopologyConnection FormsConnections On ManifoldsGeometric Structures On ManifoldsTangent Bundles
The non‑parametric version of Information Geometry has been developed in recent years, with the first basic result being the construction of a manifold structure on ℳ μ, the maximal statistical models associated to an arbitrary measure μ. The study demonstrates that the pretangent and tangent bundles on ℳ μ serve as natural domains for the mixture and exponential connections, defines a generalized Amari embedding into an Orlicz space sphere, and shows that α‑connections are defined on an α‑bundle over ℳ μ and correspond to the pull‑back of this embedding. The authors construct the manifold structure on ℳ μ, then use it to identify the pretangent and tangent bundles as domains for the connections, define the Amari embedding A Φ :ℳ μ →S Φ, and formulate the α‑connections on the α‑bundle ℱ α over ℳ μ as pull‑backs of the embedding A α :ℳ μ →S 2/1‑α. Reference 48.
The non-parametric version of Information Geometry has been developed in recent years. The first basic result was the construction of the manifold structure on ℳ μ , the maximal statistical models associated to an arbitrary measure μ (see Ref. 48). Using this construction we first show in this paper that the pretangent and the tangent bundles on ℳ μ are the natural domains for the mixture connection and for its dual, the exponential connection. Second we show how to define a generalized Amari embedding A Φ :ℳ μ →S Φ from the Exponential Statistical Manifold (ESM) ℳ μ to the unit sphere S Φ of an arbitrary Orlicz space L Φ . Finally we show that, in the non-parametric case, the α-connections ∇ α (α∈(-1,1)) must be defined on a suitable α-bundle ℱ α over ℳ μ and that the bundle-connection pair (ℱ α , ∇ α ) is simply (isomorphic to) the pull-back of the Amari embedding A α : ℳ μ →S 2/1-α were the unit sphere S 2/1-α cL 2/1-α is equipped with the natural connection.
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