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A maximum likelihood approach to density estimation with semidefinite programming.

Tadayoshi Fushiki1, Shingo Horiuchi, Takashi Tsuchiya

  • 1Institute of Statistical Mathematics, Minato-ku, Tokyo 106-8569, Japan. fushiki@ism.ac.jp

Neural Computation
|September 27, 2006
PubMed
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This study introduces a novel parametric approach for density estimation using semidefinite programming (SDP). The method efficiently computes maximum likelihood estimates for flexible density models, applicable to machine learning and statistics.

Area of Science:

  • Statistics
  • Machine Learning
  • Pattern Recognition

Background:

  • Density estimation is crucial for pattern recognition, machine learning, and statistics.
  • Existing methods may lack flexibility or computational efficiency for complex density models.

Purpose of the Study:

  • To develop a parametric density estimation approach leveraging semidefinite programming (SDP).
  • To enable rigorous maximum likelihood estimation for flexible density models.

Main Methods:

  • A density model is constructed as a product of a nonnegative polynomial and a base density (e.g., normal, exponential, uniform).
  • Maximum likelihood estimation is formulated as a variant of SDP, solvable efficiently using interior-point methods.
  • The approach handles conditions like symmetry and unimodality, with model selection via AIC.

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Main Results:

  • The proposed SDP-based method allows for polynomial-time computation of rigorous maximum likelihood estimates.
  • Demonstrated flexibility and performance through various applications, including mixture models.
  • Extended applicability shown through maximum likelihood estimation of nonstationary Poisson process intensity functions.

Conclusions:

  • The SDP-based parametric approach offers a computationally efficient and flexible framework for density estimation.
  • The method successfully estimates densities with specific properties and extends to related problems like Poisson process intensity estimation.