Weak convergence and optimal scaling of random walk Metropolis algorithms

TLDR

The study investigates how to scale the proposal distribution of a multidimensional random walk Metropolis algorithm to maximize efficiency and discusses extensions of the result. The authors analyze the algorithm by proving a weak convergence result for target densities with symmetric product form, scaling the proposal variance appropriately with dimension. They show that as dimension grows, the first component of the chain converges to a Langevin diffusion, enabling an efficiency maximization that yields an asymptotically optimal acceptance rate of 0.234 under general conditions.

Abstract

This paper considers the problem of scaling the proposal distribution of a multidimensional random walk Metropolis algorithm in order to maximize the efficiency of the algorithm. The main result is a weak convergence result as the dimension of a sequence of target densities, n, converges to $\infty$. When the proposal variance is appropriately scaled according to n, the sequence of stochastic processes formed by the first component of each Markov chain converges to the appropriate limiting Langevin diffusion process. The limiting diffusion approximation admits a straightforward efficiency maximization problem, and the resulting asymptotically optimal policy is related to the asymptotic acceptance rate of proposed moves for the algorithm. The asymptotically optimal acceptance rate is 0.234 under quite general conditions. The main result is proved in the case where the target density has a symmetric product form. Extensions of the result are discussed.