Langevin type limiting processes for adaptive MCMC

Adaptive Markov Chain Monte Carlo (AMCMC) is a class of MCMC algorithms where the proposal distribution changes at every iteration of the chain. In this case it is important to verify that such a Markov Chain indeed has a stationary distribution. In this paper we discuss a diffusion approximation to a discrete time AMCMC. This diffusion approximation is different when compared to the diffusion approximation as in Gelman et al. [5] where the state space increases in dimension to ∞. In our approach the time parameter is sped up in such a way that the limiting process (as the mesh size goes to 0) approaches to a non-trivial diffusion process.