Stochastic Complexities of Gaussian Mixtures in Variational Bayesian Approximation

Bayesian learning has been widely used and proved to be effective in many data modeling problems. However, computations involved in it require huge costs and generally cannot be performed exactly. The variational Bayesian approach, proposed as an approximation of Bayesian learning, has provided computational tractability and good generalization performance in many applications. The properties and capabilities of variational Bayesian learning itself have not been clarified yet. It is still unknown how good approximation the variational Bayesian approach can achieve. In this paper, we discuss variational Bayesian learning of Gaussian mixture models and derive upper and lower bounds of variational stochastic complexities. The variational stochastic complexity, which corresponds to the minimum variational free energy and a lower bound of the Bayesian evidence, 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 true Bayesian learning.