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An Information Quantity in Pure State Models.

Entropy (Basel, Switzerland)·2022
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Information-Geometric Approach for a One-Sided Truncated Exponential Family.

Masaki Yoshioka1, Fuyuhiko Tanaka2

  • 1Graduate School of Engineering Science, Osaka University, Osaka 560-8531, Japan.

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|May 27, 2023
PubMed
Summary

This study introduces a Riemannian metric for one-sided truncated exponential families (oTEF), a non-regular statistical model. The research reveals oTEFs possess an α = 1 parallel prior distribution and specific submodels exhibit constant negative scalar curvature.

Keywords:
alpha-parallel priorcommon scale parameterinformation geometrynon-regular modelstatistical manifoldtruncated exponential family

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Area of Science:

  • Information Geometry
  • Statistical Theory
  • Differential Geometry

Background:

  • Extensive research connects differential geometric structures (Fisher metric, α-connection) with regular statistical models.
  • Information geometry for non-regular statistical models remains understudied.
  • One-sided truncated exponential families (oTEF) represent a class of non-regular models.

Purpose of the Study:

  • To develop information geometric tools for non-regular statistical models, specifically oTEF.
  • To establish a Riemannian metric for oTEF.
  • To investigate the geometric properties of oTEF and related submodels.

Main Methods:

  • Utilizing asymptotic properties of maximum likelihood estimators.
  • Applying differential geometric concepts to statistical models.
  • Analyzing submodels including the Pareto family.

Main Results:

  • A novel Riemannian metric is provided for the one-sided truncated exponential family (oTEF).
  • The oTEF is shown to possess an α = 1 parallel prior distribution.
  • A specific submodel, including the Pareto family, demonstrates a constant negative scalar curvature.

Conclusions:

  • The study extends information geometry to non-regular models like oTEF.
  • The findings provide a foundation for further research into the geometric properties of non-regular statistical models.
  • The established metric and identified properties offer new insights into the structure of oTEF and related families.