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Nonparametric Bayesian density estimation on manifolds with applications to planar shapes.

Abhishek Bhattacharya1, David B Dunson

  • 1Department of Statistical Science, Box 90251 , Duke University , Durham, North Carolina 27708-0251 , U.S.A. ab216@stat.duke.edu dunson@duke.edu.

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Summary

This study introduces novel mixture models for statistical shape analysis on non-Euclidean spaces. These methods improve density estimation and classification accuracy in morphometrics and machine vision.

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Area of Science:

  • Statistical Shape Analysis
  • Computational Geometry
  • Non-Euclidean Statistics

Background:

  • Shape spaces, often non-Euclidean quotient manifolds, are crucial in morphometrics, medical diagnostics, and machine vision.
  • Nonparametric inference on these spaces typically involves estimating centers and spreads, but full likelihood methods offer enhanced density estimation capabilities.

Purpose of the Study:

  • To propose a flexible class of mixture models for nonparametric density estimation on compact metric spaces, with a focus on planar shape spaces.
  • To investigate the theoretical properties of these models, including Kullback-Leibler properties for large support and weak posterior consistency, using a Bayesian approach with nonparametric priors.

Main Methods:

  • Construction of mixture models with suitable kernels on general compact metric spaces and specifically on planar shape spaces.
  • Application of a Bayesian framework with nonparametric priors on the mixing distribution.
  • Development of Gibbs sampling methods for posterior computation.
  • Utilizing shape-based predictors for density estimation and classification tasks.

Main Results:

  • Established conditions for the Kullback-Leibler property, ensuring desirable theoretical properties of the proposed models.
  • Demonstrated the effectiveness of the developed Gibbs sampling methods for posterior computation.
  • Simulation studies indicated superior estimation performance compared to existing statistical shape analysis approaches.

Conclusions:

  • The proposed mixture models provide a powerful and theoretically sound framework for statistical analysis on shape spaces.
  • The methods offer improved performance in density estimation and classification, advancing applications in morphometrics and machine vision.
  • This work facilitates more robust and accurate statistical inferences in complex shape analysis problems.