Stochastic Complexity for Mixture of Exponential Families in Variational Bayes

Authors: Kazuho Watanabe and Sumio Watanabe

Source: Algorithmic Learning Theory, 16th International Conference, ALT 2005, Singapore, October 2005, Proceedings, (Sanjay Jain, Hans Ulrich Simon and Etsuji Tomita, Eds.), Lecture Notes in Artificial Intelligence 3734, pp. 107 - 121, Springer 2005.

Abstract. The Variational Bayesian learning, proposed as an approximation of the Bayesian learning, has provided computational tractability and good generalization performance in many applications. However, little has been done to investigate its theoretical properties.

In this paper, we discuss the Variational Bayesian learning of the mixture of exponential families and derive the asymptotic form of the stochastic complexities. We show that the stochastic complexities become smaller than those of regular statistical models, which implies the advantage of the Bayesian learning still remains in the Variational Bayesian learning. Stochastic complexity, which is called the marginal likelihood or the free energy, not only becomes important in addressing the model selection problem but also enables us to discuss the accuracy of the Variational Bayesian approach as an approximation of the true Bayesian learning.

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