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Global Geometry of Bayesian Statistics.

Atsuhide Mori1

  • 1Department of Mathematics, Osaka Dental University, Osaka 573-1121, Japan.

Entropy (Basel, Switzerland)
|December 8, 2020
PubMed
Summary
This summary is machine-generated.

This study reveals a symmetry in normal distribution information geometry, linking it to Bayesian inference and current geometry. The findings generalize previous work using symplectic foliations derived from Cholesky decomposition.

Keywords:
Cholesky decompositioncontact structurefoliationinformation geometrypoisson structuresymplectic structure

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Area of Science:

  • Information Geometry
  • Statistical Physics
  • Differential Geometry

Background:

  • A novel symmetry was previously identified in the information geometry of normal distributions.
  • This symmetry was also observed in the symplectic/contact geometry of Hilbert modular cusps.
  • A connection between contact Hamiltonian flow and Bayesian inference on normal distributions was noted.

Purpose of the Study:

  • To translate Bayesian statistics and information geometry into the framework of current geometry.
  • To generalize previously discovered symmetries in normal distribution information geometry.
  • To bridge the gap between statistical and geometrical communities.

Main Methods:

  • Describing Bayesian statistics and information geometry using current geometric language.
  • Foliating the space of multivariate normal distributions into symplectic leaves.
  • Utilizing the Cholesky decomposition of covariance matrices to define the foliation.

Main Results:

  • A generalized symmetry in the information geometry of normal distributions was established.
  • The space of multivariate normal distributions was successfully foliated by symplectic leaves.
  • The Cholesky decomposition was shown to be fundamental to this geometric structure.

Conclusions:

  • The study successfully integrates Bayesian statistics and information geometry with current geometric concepts.
  • The generalized symmetry provides a new perspective on the geometric underpinnings of statistical inference.
  • This work aims to foster interdisciplinary interest and collaboration between geometers and statisticians.