  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 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.
