CONNECTIONS ON STATISTICAL MANIFOLDS OF DENSITY OPERATORS BY GEOMETRY OF NONCOMMUTATIVE L^p-SPACES

Abstract

Let N be a statistical manifold of density operators, with respect to an n.s.f. trace τ on a semifinite von Neumann algebra M . If S p is the unit sphere of the noncommutative space L p (M, τ ), using the noncommutative Amari embedding ρ ∈ N → ρ 1/p ∈ S p , we define a noncommutative α-bundle-connection pair (F α , ∇ α ), by the pullback technique. In the commutative case we show that it coincides with the construction of nonparametric Amari-Čentsov α-connection made in Ref. 8 by Gibilisco and Pistone.

Information Geometry Formalism for the Spatially Homogeneous Boltzmann Equation

“…under the normal collision distribution is obtained from Equation (11). 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.…”

“…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.…”

Examples of the Application of Nonparametric Information Geometry to Statistical Physics

“…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).…”