Langevin type limiting processes for Adaptive MCMC

TL;DR

This paper develops a diffusion approximation for adaptive MCMC algorithms, providing insights into their limiting behavior by analyzing a continuous-time diffusion process different from traditional high-dimensional limits.

Contribution

It introduces a novel diffusion approximation for adaptive MCMC, focusing on a continuous-time limit rather than increasing state space dimension.

Findings

Diffusion process converges to a non-trivial continuous-time limit.

Provides theoretical foundation for understanding adaptive MCMC dynamics.

Differentiates from previous high-dimensional diffusion approximations.

Abstract

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, Gilks and Roberts (1997) where the state space increases in dimension to infinity. In our approach the time parameter is sped up in such a way that the limiting distribution (as the mesh size goes to 0) is to a non-trivial continuous time diffusion process.