Diffusion Approximation to Adaptive MCMC

Adaptive MCMC algorithms are designed to tune the chain properly to decrease the time to convergence to stationary. However since the transition kernel is changed at each step of iteration proper ergodicity conditions has to be ensured. Here we define a discrete time Adaptive MCMC algorithm and study its convergence properties. We apply the diffusion approximation method to a discrete time Adaptive MCMC procedure to obtain the limiting stochastic differential equation governing the dynamics of the adaptation parameter $\theta$ and the state space variable $X$. The solution to the coupled equation will give the stationary distribution of the chain. Comparison of rates of convergence between the standard and the adaptive method will be of interest.