A kernel mixing strategy for use in adaptive Markov chain Monte Carlo and stochastic optimization contexts

Abstract

Performing Markov chain Monte Carlo parameter estimation on complex mathematical models can quickly lead to endless searching through highly multimodal parameter spaces. For computationally complex models, one rarely has prior knowledge of the optimal proposal distribution. In such cases, the Markov chain can become trapped near a suboptimal mode, lowering the computational efficiency of the method. With these challenges in mind, we present a novel MCMC kernel which incorporates both mixing and adaptation. The method is flexible and robust enough to handle parameter spaces that are highly multimodal. Other advantages include not having to locate a near-optimal mode with a different method beforehand, as well as requiring minimal computational and storage overhead from standard Metropolis. Additionally, it can be applied in any stochastic optimization context which uses a Gaussian kernel. We provide results from several benchmark problems, comparing the kernel's performance in both optimization and MCMC cases. For the former, we incorporate the kernel into a simulated annealing method and real-coded genetic algorithm. For the latter, we incorporate it into the standard Metropolis and adaptive Metropolis methods.