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A Geometric Approach to Average Problems on Multinomial and Negative Multinomial Models.

Mingming Li1,2, Huafei Sun1,2, Didong Li3

  • 1School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China.

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

This study explores geometric structures within multinomial and negative multinomial models, revealing parallel results for average computations like midpoints and Karcher means.

Keywords:
Karcher meanaverage problemgeometric midpointsstructure characterization

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Area of Science:

  • Statistics
  • Probability Theory
  • Information Geometry

Background:

  • Multinomial and negative multinomial models are fundamental in statistical analysis.
  • Understanding their geometric properties is crucial for advanced statistical methods.
  • Existing research often treats these models separately, limiting comparative insights.

Purpose of the Study:

  • To investigate the complementary geometric structures of multinomial and negative multinomial models.
  • To formulate and compute average problems within these geometric frameworks.
  • To establish parallel results for averaging techniques across both models.

Main Methods:

  • Derivation of fundamental geometric quantities: Fisher-Riemannian metrics, α-connections, and α-curvatures.
  • Application of geometric structures to define and compute average methods.
  • Focus on calculating the midpoint of two points and the Karcher mean of multiple points.

Main Results:

  • Demonstration of complementary geometric structures inherent to the multinomial and negative multinomial models.
  • Successful formulation and computation of average problems leveraging these geometric properties.
  • Identification of analogous results for averaging techniques in both model types.

Conclusions:

  • The geometric approach provides a unified framework for analyzing average problems in related statistical models.
  • Parallel results highlight deep connections between multinomial and negative multinomial distributions.
  • This research offers new tools for statistical inference and data analysis in these models.