Adaptive Markov chain Monte Carlo for auxiliary variable method and its application to parallel tempering

Auxiliary variable methods such as the Parallel Tempering and the cluster Monte Carlo methods generate samples that follow a target distribution by using proposal and auxiliary distributions. In sampling from complex distributions, these algorithms are highly more efficient than the standard Markov chain Monte Carlo methods. However, their performance strongly depends on their parameters and determining the parameters is critical. In this paper, we proposed an algorithm for adapting the parameters during drawing samples and proved the convergence theorem of the adaptive algorithm. We applied our algorithm to the Parallel Tempering. That is, we developed an adaptive Parallel Tempering that tunes the parameters on the fly. We confirmed the effectiveness of our algorithm through the validation of the adaptive Parallel Tempering, comparing samples from the target distribution by the adaptive Parallel Tempering and samples by conventional algorithms.