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Integration of Stochastic Models by Minimizing -Divergence

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

When there are a number of stochastic models in the form of probability distributions, one needs to integrate them. Mixtures of distributions are frequently used, but exponential mixtures also provide a good means of integration. This letter proposes a one-parameter family of integration, called -integration, which includes all of these well-known integrations. These are generalizations of various averages of numbers such as arithmetic, geometric, and harmonic averages. There are psychophysical experiments that suggest that -integrations are used in the brain. The -divergence between two distributions is defined, which is a natural generalization of Kullback-Leibler divergence and Hellinger distance, and it is proved that -integration is optimal in the sense of minimizing -divergence. The theory is applied to generalize the mixture of experts and the product of experts to the -mixture of experts. The -predictive distribution is also stated in the Bayesian framework.