Information geometry in a global setting
type: Item
instance of: scholarly article
title: Information geometry in a global setting
zbMATH Open document ID: 1411.57041
author: Atsuhide Mori
published in: Hiroshima Mathematical Journal
publication date: 2019-03-05T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.hmj/1544238029
review text: Information geometry is a branch of statistical methods that apply techniques of differential geometry to the study of a parameter space of a family of probability distributions of a statistical model which can be viewed as points on a Riemannian manifold. \par A mapping torus $T_A$ of a hyperbolic automorphism $A\in \mathrm{SL}(2;\mathbb Z)$ of the 2-torus $\mathbb T^2$ is called a $\mathrm{Sol}^3$-manifold. The suspension Anosov flow on $T_A$ induces a positive contact structure $\xi_+=\ker(\beta_++\beta_-)$ and a negative contact structure $\xi_-=\ker(\beta_+-\beta_-)$, where $\beta_+$ and $\beta_-$ are the 1-forms defining the Anosov foliations. They form a bi-contact structure on $T_A$. If the parameter space is the upper half-plane $\mathbb H=\{(m,s): s>0\}\subset\mathbb R^2$ with a certain non-Kähler metric and a non-metric connection, where $m$ denotes the mean and $s$ is the standard deviation, then the local structures are derived from a certain global function $D$ on $\mathbb H\times\mathbb H$ which is fundamental in information theory. On the other hand, the product $\mathbb H\times\mathbb H$ parametrizes abelian surfaces in moduli theory, that is, the quotient of the product $\mathbb H\times\mathbb H$ under the action of a Hilbert modular group is a non-compact singular complex surface which parametrizes the isomorphism classes of complex abelian surfaces with a real multiplication structure. Its topology is defined by describing each end of a Hilbert modular surface as a cusp, namely the neighborhood of $\infty$ of the quotient $M$ of $\mathbb H\times\mathbb H$ under a certain action of a semi-direct product $\mathbb Z\ltimes\mathbb Z^2$ of lattices of $\mathbb R$ and $\mathbb R^2$. From the contact topological point of view, $M\cup \{\infty\}$ is the cone with positive and negative symplectic structures whose base $N$ is a $\mathrm{Sol}^3$-manifold with the positive and negative contact structures $\xi_{\pm}$ associated to the suspension Anosov flow. …
reviewed by: Andrew Bucki
Mathematics Subject Classification ID: 57R17
Mathematics Subject Classification ID: 57R30
Mathematics Subject Classification ID: 62B10
Mathematics Subject Classification ID: 53C12
zbMATH DE Number: 7032359
zbMATH Keywords: information geometry
zbMATH Keywords: contact structure
zbMATH Keywords: foliation
MaRDI profile type: Publication
OpenAlex ID: W2905566736
Wikidata QID: Q128763435
DOI: 10.32917/HMJ/1544238029
description: scientific article; zbMATH DE number 7032359