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Geometry of Statistical Manifolds.

Paul W Vos1

  • 1Department of Public Health, Brody School of Medicine, East Carolina University, Greenville, NC 27834, USA.

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PubMed
Summary
This summary is machine-generated.

Statistical manifolds extend Riemannian geometry to probability distributions. This study generalizes point estimates to functions, defining Λ-information bounded by Fisher information, with applications in the two-sample problem.

Keywords:
Hilbert bundlegeneralized estimationinformationnuisance parametersorthogonalizationparameter-invarianceslope

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Area of Science:

  • * Differential Geometry
  • * Statistical Inference
  • * Information Theory

Background:

  • * Statistical manifolds are Riemannian manifolds where points represent probability distributions.
  • * Traditional point estimates are single points in a manifold; this work generalizes them to functions.
  • * The Fisher information metric is a key concept in the geometry of statistical manifolds.

Purpose of the Study:

  • * To generalize the concept of point estimates to functions on statistical manifolds.
  • * To introduce and define Λ-information based on the geometric properties of these generalized estimators.
  • * To explore the relationship between Λ-information and Fisher information.

Main Methods:

  • * Extension of the tangent bundle (TM) to a Hilbert bundle (HM) to accommodate distribution-valued estimators.
  • * Definition of generalized estimators (gθ^) as functions over parameter spaces.
  • * Utilizing geometric properties, specifically expected slopes, to define Λ-information.

Main Results:

  • * Generalized estimators (gθ^) are characterized by their geometric properties.
  • * Λ-information is defined based on the expected slopes of generalized estimators.
  • * Fisher information (I) provides an upper bound for Λ-information: Λ(g) ≤ I.

Conclusions:

  • * A geometric framework for understanding generalized statistical estimators has been established.
  • * The Λ-information offers a new measure related to statistical models and data.
  • * The utility of this geometric perspective is demonstrated through the two-sample problem.