Statistical Einstein manifolds of exponential families with group-invariant potential functions

Linyu Peng, Zhenning Zhang

Research output: Contribution to journalArticle

Abstract

This paper mainly contributes to a classification of statistical Einstein manifolds, namely statistical manifolds at the same time are Einstein manifolds. A statistical manifold is a Riemannian manifold, each of whose points is a probability distribution. With the Fisher information metric as a Riemannian metric, information geometry was developed to understand the intrinsic properties of statistical models, which play important roles in statistical inference, etc. Among all these models, exponential families is one of the most important kinds, whose geometric structures are fully determined by their potential functions. To classify statistical Einstein manifolds, we derive partial differential equations for potential functions of exponential families; special solutions of these equations are obtained through the ansatz method as well as group-invariant solutions via reductions using Lie point symmetries.

Original languageEnglish
Pages (from-to)2104-2118
Number of pages15
JournalJournal of Mathematical Analysis and Applications
Volume479
Issue number2
DOIs
Publication statusPublished - 2019 Nov 15

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Statistical Manifold
Einstein Manifold
Exponential Family
Potential Function
Invariant
Probability distributions
Partial differential equations
Group Invariant Solutions
Information Geometry
Lie Point Symmetries
Fisher Information
Geometry
Geometric Structure
Riemannian Metric
Statistical Inference
Statistical Model
Riemannian Manifold
Probability Distribution
Partial differential equation
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Keywords

  • Einstein manifold
  • Group-invariant solutions
  • Information geometry
  • Symmetry reduction

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

Cite this

Statistical Einstein manifolds of exponential families with group-invariant potential functions. / Peng, Linyu; Zhang, Zhenning.

In: Journal of Mathematical Analysis and Applications, Vol. 479, No. 2, 15.11.2019, p. 2104-2118.

Research output: Contribution to journalArticle

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