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8 pages, 296 KiB

Open AccessProceeding Paper

Dynamical Systems over Lie Groups Associated with Statistical Transformation Models

by Daisuke Tarama and Jean-Pierre Françoise

Phys. Sci. Forum 2022, 5(1), 21; https://doi.org/10.3390/psf2022005021 - 7 Dec 2022

Viewed by 1360

A statistical transformation model consists of a smooth data manifold, on which a Lie group smoothly acts, together with a family of probability density functions on the data manifold parametrized by elements in the Lie group. For such a statistical transformation model, the Fisher–Rao semi-definite metric and the Amari–Chentsov cubic tensor are defined in the Lie group. If the family of probability density functions is invariant with respect to the Lie group action, the Fisher–Rao semi-definite metric and the Amari–Chentsov tensor are left-invariant, and hence we have a left-invariant structure of a statistical manifold. In the present work, the general framework of statistical transformation models is explained. Then, the left-invariant geodesic flow associated with the Fisher–Rao metric is considered for two specific families of probability density functions on the Lie group. The corresponding Euler–Poincaré and the Lie–Poisson equations are explicitly found in view of geometric mechanics. Related dynamical systems over Lie groups are also mentioned. A generalization in relation to the invariance of the family of probability density functions is further studied. Full article

(This article belongs to the Proceedings of The 41st International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering)

30 pages, 460 KiB

Open AccessArticle

Differential Geometric Aspects of Parametric Estimation Theory for States on Finite-Dimensional C^∗-Algebras

by Florio M. Ciaglia, Jürgen Jost and Lorenz Schwachhöfer

Entropy 2020, 22(11), 1332; https://doi.org/10.3390/e22111332 - 23 Nov 2020

Cited by 7 | Viewed by 2779

Abstract

A geometrical formulation of estimation theory for finite-dimensional

C^{*}

-algebras is presented. This formulation allows to deal with the classical and quantum case in a single, unifying mathematical framework. The derivation of the Cramer–Rao and Helstrom bounds for parametric statistical models with [...] Read more.

A geometrical formulation of estimation theory for finite-dimensional

C^{*}

(This article belongs to the Special Issue Quantum Statistical Decision and Estimation Theory)

61 pages, 2469 KiB

Open AccessReview

An Elementary Introduction to Information Geometry

by Frank Nielsen

Entropy 2020, 22(10), 1100; https://doi.org/10.3390/e22101100 - 29 Sep 2020

Cited by 86 | Viewed by 44276

Abstract

In this survey, we describe the fundamental differential-geometric structures of information manifolds, state the fundamental theorem of information geometry, and illustrate some use cases of these information manifolds in information sciences. The exposition is self-contained by concisely introducing the necessary concepts of differential [...] Read more.

(This article belongs to the Special Issue Review Papers for Entropy)

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27 pages, 441 KiB

Open AccessArticle

From the Jordan Product to Riemannian Geometries on Classical and Quantum States

by Florio M. Ciaglia, Jürgen Jost and Lorenz Schwachhöfer

Entropy 2020, 22(6), 637; https://doi.org/10.3390/e22060637 - 8 Jun 2020

Cited by 15 | Viewed by 2370

Abstract

The Jordan product on the self-adjoint part of a finite-dimensional

C^{*}

-algebra

A

is shown to give rise to Riemannian metric tensors on suitable manifolds of states on

A

, and the covariant derivative, the geodesics, the Riemann tensor, and the sectional [...] Read more.

The Jordan product on the self-adjoint part of a finite-dimensional

C^{*}

-algebra

A

is shown to give rise to Riemannian metric tensors on suitable manifolds of states on

A

, and the covariant derivative, the geodesics, the Riemann tensor, and the sectional curvature of all these metric tensors are explicitly computed. In particular, it is proved that the Fisher–Rao metric tensor is recovered in the Abelian case, that the Fubini–Study metric tensor is recovered when we consider pure states on the algebra

B (H)

of linear operators on a finite-dimensional Hilbert space

H

, and that the Bures–Helstrom metric tensors is recovered when we consider faithful states on

B (H)

. Moreover, an alternative derivation of these Riemannian metric tensors in terms of the GNS construction associated to a state is presented. In the case of pure and faithful states on

B (H)

, this alternative geometrical description clarifies the analogy between the Fubini–Study and the Bures–Helstrom metric tensor. Full article

(This article belongs to the Section Quantum Information)

273 KiB

Open AccessArticle

On Monotone Embedding in Information Geometry

by Jun Zhang

Entropy 2015, 17(7), 4485-4499; https://doi.org/10.3390/e17074485 - 25 Jun 2015

Cited by 19 | Viewed by 4706

Abstract

A paper was published (Harsha and Subrahamanian Moosath, 2014) in which the authors claimed to have discovered an extension to Amari's \(\alpha\)-geometry through a general monotone embedding function. It will be pointed out here that this so-called \((F, G)\)-geometry (which includes \(F\)-geometry as a special case) is identical to Zhang's (2004) extension to the \(\alpha\)-geometry, where the name of the pair of monotone embedding functions \(\rho\) and \(\tau\) were used instead of \(F\) and \(H\) used in Harsha and Subrahamanian Moosath (2014). Their weighting function \(G\) for the Riemannian metric appears cosmetically due to a rewrite of the score function in log-representation as opposed to \((\rho, \tau)\)-representation in Zhang (2004). It is further shown here that the resulting metric and \(\alpha\)-connections obtained by Zhang (2004) through arbitrary monotone embeddings is a unique extension of the \(\alpha\)-geometric structure. As a special case, Naudts' (2004) \(\phi\)-logarithm embedding (using the so-called \(\log_\phi\) function) is recovered with the identification \(\rho=\phi, \, \tau=\log_\phi\), with \(\phi\)-exponential \(\exp_\phi\) given by the associated convex function linking the two representations. Full article

(This article belongs to the Special Issue Information, Entropy and Their Geometric Structures)

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