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Stochastic complexities of general mixture models in variational Bayesian learning.

Kazuho Watanabe1, Sumio Watanabe

  • 1Department of Computational Intelligence and Systems Science, Tokyo Institute of Technology, Japan. kazuho23@pi.titech.ac.jp

Neural Networks : the Official Journal of the International Neural Network Society
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This study analyzes variational Bayesian learning for mixture models, deriving its asymptotic stochastic complexity. Results show variational Bayesian methods retain advantages of true Bayesian learning and clarify hyperparameter influence.

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Area of Science:

  • Machine Learning
  • Statistical Inference
  • Computational Statistics

Background:

  • Variational Bayesian (VB) learning approximates Bayesian learning, offering computational advantages.
  • Despite widespread application, the theoretical properties of VB learning remain underexplored.
  • General mixture models are fundamental in statistical modeling and pattern recognition.

Purpose of the Study:

  • To investigate the theoretical properties of variational Bayesian learning for general mixture models.
  • To derive the asymptotic form of stochastic complexity for VB learning of exponential-family mixtures.
  • To analyze the impact of hyperparameters and approximation accuracy in VB learning.

Main Methods:

  • Derivation of the asymptotic stochastic complexity (free energy) for variational Bayesian inference.
  • Analysis of mixture models based on exponential-family distributions.
  • Theoretical investigation of approximation bounds and hyperparameter influence.

Main Results:

  • The asymptotic stochastic complexity for VB learning of mixture models was obtained.
  • Stochastic complexities in VB learning were found to be smaller than in regular statistical models.
  • Derived bounds quantify the influence of hyperparameters and the accuracy of the VB approximation.

Conclusions:

  • Variational Bayesian learning effectively retains the advantages of true Bayesian learning.
  • The derived theoretical results provide insights into the behavior and performance of VB methods.
  • This work contributes to a deeper understanding of the theoretical underpinnings of variational Bayesian inference.