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-mu the maximal statistical models associated to an arbitrary measure mu (see Ref. 48). Using this construction we first show in this paper that the pretangent and the tangent bundles on M-mu 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(Phi): M-mu --> S-Phi from the Exponential Statistical Manifold (ESM) M-mu to the unit sphere S-Phi of an arbitrary Orlicz space L-Phi. Finally we show that, in the non-parametric case, the cr-connections del(alpha) (alpha is an element of (-1, 1)) must be defined on a suitable alpha-bundle F-alpha over M-mu and that the bundle-connection pair (F-alpha,del(alpha)) is simply (isomorphic to) the pull-back of the Amari embedding A(alpha): M-mu --> S2/1-alpha where the unit sphere S(2/1-alpha)cL(2/1-alpha) is equipped with the natural connection.