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Integration of stochastic models by minimizing alpha-divergence.

Shun-ichi Amari¹

¹RIKEN Brain Science Institute, Wako-shi, Hirosawa 2-1, Saitama 351-0198, Japan. amari@brain.riken.jp

Neural Computation

|August 25, 2007

Summary

This study introduces alpha-integration, a novel method unifying various probability distribution integrations and averages. Alpha-integration is proven optimal for minimizing alpha-divergence, with potential brain applications.

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Area of Science:

Information Theory
Probability Theory
Machine Learning

Background:

Integrating multiple probability distributions is crucial for stochastic modeling.
Existing methods like exponential mixtures offer integration solutions.
Generalizations of averages like arithmetic, geometric, and harmonic averages are widely used.

Purpose of the Study:

To propose a unified, one-parameter family of integration called alpha-integration.
To define alpha-divergence as a generalization of Kullback-Leibler divergence and Hellinger distance.
To explore the neurobiological plausibility and applications in machine learning.

Main Methods:

Introduced a one-parameter family of integration: alpha-integration.
Defined alpha-divergence between probability distributions.
Proved alpha-integration's optimality in minimizing alpha-divergence.
Generalized mixture of experts and product of experts models.

Main Results:

Alpha-integration encompasses existing integration methods and averages.
Alpha-divergence generalizes established divergence measures.
Alpha-integration is shown to be optimal for minimizing alpha-divergence.
Developed alpha-mixture of experts and alpha-predictive distributions.

Conclusions:

Alpha-integration provides a unified framework for distribution integration.
The proposed method has theoretical optimality and practical applications in machine learning.
Psychophysical evidence suggests potential use in neural processing.