Nonparametric Information Geometry

Abstract

Abstract. The differential-geometric structure of the set of positive densities on a given measure space has raised the interest of many mathematicians after the discovery by C.R. Rao of the geometric meaning of the Fisher information. Most of the research is focused on parametric statistical models. In series of papers by author and coworkers a particular version of the nonparametric case has been discussed. It consists of a minimalistic structure modeled according the theory of exponential families: given a reference density other densities are represented by the centered log likelihood which is an element of an Orlicz space. This mappings give a system of charts of a Banach manifold. It has been observed that, while the construction is natural, the practical applicability is limited by the technical difficulty to deal with such a class of Banach spaces. It has been suggested recently to replace the exponential function with other functions with similar behavior but polynomial growth at infinity in order to obtain more tractable Banach spaces, e.g. Hilbert spaces. We give first a review of our theory with special emphasis on the specific issues of the infinite dimensional setting. In a second part we discuss two specific topics, differential equations and the metric connection. The position of this line of research with respect to other approaches is briefly discussed.

Information Geometry Formalism for the Spatially Homogeneous Boltzmann Equation

“…We retell the basic story in order to introduce our notations and the IG background. Orlicz spaces as a setting for Boltzmann's equation have been recently proposed by [33], while the use of exponential statistical manifolds has been suggested in [18,Example 11] and sketched in [19,Section 4.4]. We start with an improvement of the latter, a few repetitions being justified by consistency between this presentation and [32, Sections 1.3, 4.5-6], compare also Proposition 10 below.…”

“…We refer to [18,19] for further details on the geometric structure, namely the Hilbert bundle, the tangent mapping of an homeomorphism, the Riemannian Hessian. We now turn to a basic example.…”

“…In fact Π is the projector on the subspace generated by κ − σ where (v, w, σ) → κ = v − w is uniformly distributed and independent from σ. Hence, Equation (18) shows that (v, w, σ) → κ − σ is uniformly distributed on S 2 so that the distribution in Equation (24) is the ν measure defined in Equation (13). Conditionally to Π, the normal collision distribution is Gaussian with covariance…”

“…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.…”