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) --> S(2/1-alpha) where the unit sphere S(2/1-alpha)cL(2/1-alpha) is equipped with the natural connection.