Connections on Non-Parametric Statistical Manifolds by Orlicz Space Geometry

Abstract

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 ℳμ, 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 ℳμ 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 AΦ:ℳμ→SΦ from the Exponential Statistical Manifold (ESM) ℳμ to the unit sphere SΦ of an arbitrary Orlicz space LΦ. Finally we show that, in the non-parametric case, the α-connections ∇α(α∈(-1,1)) must be defined on a suitable α-bundle ℱα over ℳμ and that the bundle-connection pair (ℱα, ∇α) is simply (isomorphic to) the pull-back of the Amari embedding Aα: ℳμ→S2/1-α were the unit sphere S2/1-αcL2/1-α is equipped with the natural connection.

Examples of the Application of Nonparametric Information Geometry to Statistical Physics

“…Our results on parallel transports and connections are a development, not yet complete, of previous work on statistical bundles in [6,8,14,23].…”

“…is provided with an atlas of charts by using the isometries, U q p : H p → H q , which result from the pull-back of the metric connection on the sphere S µ = f ∈ L 2 (µ) : f 2 dµ = 1 ; see [6,8,23] and [14] (Section 4).…”

“…The differential geometry involved in their construction is finite dimensional, and the formalism is based on coordinate systems. Following a suggestion by Phil Dawid in [2][3][4], a particular nonparametric version of the Amari-Nagaoka theory was developed in a series of papers [5][6][7][8][9][10][11][12][13][14], where the set P > of all strictly positive probability densities of a measure space is shown to be a Banach manifold (as defined in [15][16][17]) modeled on an Orlicz Banach space, see [18] (Ch II).…”

Information Geometry Formalism for the Spatially Homogeneous Boltzmann Equation

“…The basic case of a finite state space has been extended by Amari and coworkers to the case of a parametric set of strictly positive probability densities on a generic sample space. Following a suggestion by Dawid in [7,8], a particular nonparametric version of that theory was developed in a series of papers [9][10][11][12][13][14][15][16][17][18][19], where the set P > of all strictly positive probability densities of a measure space is shown to be a Banach manifold (as it is defined in [20][21][22]) modeled on an Orlicz Banach space, see, e.g., [23, Chapter II].…”

“…This material is included for convenience only and this part should be skipped by any reader aware of any of the papers [9][10][11][12][13][14][15][16][17][18][19] quoted above. The following Section 3 is mostly based on the same references and it is intended to introduce that manifold structure and to give a first example of application to the study of Kullback-Liebler divergence.…”