Comparison between Standard and Adaptive MCMC via their diffusion limits

Gopal K. Basak, Arunangshu Biswas

Published 2012-01-06, updated 2014-11-18Version 3

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 an MC indeed has a stationary distribution. In this paper we discuss a diffusion approximation to a discrete time AMCMC and that of Standard MCMC (SMCMC). This diffusion approximation is different when compared to the diffusion approximation as in Gelman, Gilks and Roberts (1997) where the state space increases in dimension to $\infty$. In our approach the time parameter is sped up in such a way that the limiting distribution is to a non-trivial continuous time diffusion process. We provide the tables and plots related to that of SMCMC and AMCMC for comparison.