Adaptive equi-energy sampler: Convergence and illustration

Markov chain Monte Carlo (MCMC) methods allow to sample a distribution known up to a multiplicative constant. Classical MCMC samplers are known to have very poor mixing properties when sampling multimodal distributions. The Equi-Energy sampler is an interacting MCMC sampler proposed by Kou, Zhou and Wong in 2006 to sample difficult multimodal distributions. This algorithm runs several chains at different temperatures in parallel, and allow lower-tempered chains to jump to a state from a higher-tempered chain having an energy "close" to that of the current state. A major drawback of this algorithm is that it depends on many design parameters and thus, requires a significant effort to tune these parameters. In this article, we introduce an Adaptive Equi-Energy (AEE) sampler that automates the choice of the selection mecanism when jumping onto a state of the higher-temperature chain. We prove the ergodicity and a strong law of large numbers for AEE, and for the original Equi-Energy sampler as well. Finally, we apply our algorithm to motif sampling in DNA sequences.