In real-world problems, the probability distribution of a given data set usually has multiple modes. The probability distribution can be estimated as a mixture of single-mode distributions. A powerful method for estimating the mixture model is Bayesian learning. However, this method is computationally intensive, so many researchers adopt calculus of variation and propose variational Bayesian learning, which is an approximated version of Bayesian learning that is computationally tractable.

Although variational Bayesian learning has been applied successfully to solve many real-world problems, its theoretical properties have not been studied extensively. This paper considers one of the performance properties, stochastic complexity, and derives its upper and lower bounds for variational Bayesian learning. The theoretical results will provide a guideline for researchers who will adopt this approach in practice.

In short, this paper is very rigorous. The mathematical derivation contributes significantly to the areas of artificial intelligence and statistics.