Connections on Non-Parametric Statistical Manifolds by Orlicz Space Geometry

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-αcL^2/1-α is equipped with the natural connection.