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A Simple Approximation Method for the Fisher-Rao Distance between Multivariate Normal Distributions.

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This study introduces a novel method to approximate Fisher-Rao distances between multivariate normal distributions using curve discretization and Jeffreys divergence. The approach offers a computationally efficient way to measure differences between complex probability distributions.

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Fisher–Rao normal manifoldelliptical distributionexponential familyinformation geometryisometric embeddingmaximal invariantsymmetric positive–definite matrix cone

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Area of Science:

  • Information Geometry
  • Statistical Inference
  • Computational Statistics

Background:

  • The Fisher-Rao distance is a fundamental metric in information geometry for measuring the difference between probability distributions.
  • Approximating this distance for multivariate normal distributions is computationally challenging, especially in higher dimensions.

Purpose of the Study:

  • To develop a simple and efficient method for approximating the Fisher-Rao distance between multivariate normal distributions.
  • To evaluate the accuracy of the proposed approximation technique.

Main Methods:

  • Discretizing curves connecting multivariate normal distributions.
  • Approximating Fisher-Rao distances using the square roots of Jeffreys divergences between successive distributions.
  • Comparing linear interpolation curves with a curve from Calvo and Oller's isometric embedding.

Main Results:

  • The proposed method provides a viable approximation for the Fisher-Rao distance.
  • Experimental results assess the approximation quality against lower and upper bounds.
  • Information-geometric properties of the Calvo and Oller embedding are elucidated.

Conclusions:

  • The developed technique offers a practical approach to approximating Fisher-Rao distances for multivariate normal distributions.
  • The study contributes to understanding the geometry of normal distribution manifolds and provides tools for statistical inference.