Stochastic Approximation in Monte Carlo Computation

The Wang-Landau (WL) algorithm is an adaptive Markov chain Monte Carlo algorithm used to calculate the spectral density for a physical system. A remarkable feature of the WL algorithm is that it is not trapped by local energy minima, which is very important for systems with rugged energy landscapes. This feature has led to many successful applications of the algorithm in statistical physics and biophysics; however, there does not exist rigorous theory to support its convergence, and the estimates produced by the algorithm can reach only a limited statistical accuracy. In this article we propose the stochastic approximation Monte Carlo (SAMC) algorithm, which overcomes the shortcomings of the WL algorithm. We establish a theorem concerning its convergence. The estimates produced by SAMC can be improved continuously as the simulation proceeds. SAMC also extends applications of the WL algorithm to continuum systems. The potential uses of SAMC in statistics are discussed through two classes of applications, importance sampling and model selection. The results show that SAMC can work as a general importance sampling algorithm and a model selection sampler when the model space is complex.