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Information criterion for approximation of unnormalized densities.

John Y Choe1, Yen-Chi Chen2, Nick Terry1

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This study introduces a new method for approximating unknown probability densities using a novel Cross-Entropy Information Criterion (CIC). The CIC helps select the best parametric model, improving density approximation for Bayesian inference and importance sampling.

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Area of Science:

  • Statistics
  • Computational Statistics
  • Machine Learning

Background:

  • Density approximation is crucial in Bayesian inference and importance sampling.
  • Traditional methods like AIC and BIC are insufficient for approximating densities evaluated up to a normalizing constant.
  • Parametric density approximation requires robust model selection techniques.

Purpose of the Study:

  • To develop a novel information criterion for selecting parametric density approximation models.
  • To address the limitations of traditional model selection criteria in density approximation.
  • To propose an iterative method for approximating unknown densities.

Main Methods:

  • Formulating density approximation as a model selection problem within pre-specified distribution families.
  • Minimizing cross-entropy to measure the deviation between parametric models and the target density.
  • Proposing and validating the Cross-Entropy Information Criterion (CIC) as an asymptotically unbiased estimator of cross-entropy.

Main Results:

  • The proposed Cross-Entropy Information Criterion (CIC) is shown to be an asymptotically unbiased estimator of cross-entropy.
  • An iterative method based on CIC minimization effectively approximates target densities.
  • Numerical studies demonstrate the method's ability to select well-approximating parametric models.

Conclusions:

  • The CIC provides a principled approach to model selection for density approximation.
  • The iterative CIC minimization method offers a practical solution for approximating unknown densities.
  • This work advances statistical methods for density estimation and model selection.