Optimal acceptance rates for Metropolis algorithms: Moving beyond 0.234

Recent optimal scaling theory has produced a condition for the asymptotically optimal acceptance rate of Metropolis algorithms to be the well-known 0.234 when applied to certain multi-dimensional target distributions. These d -dimensional target distributions are formed of independent components, each of which is scaled according to its own function of d . We show that when the condition is not met the limiting process of the algorithm is altered, yielding an asymptotically optimal acceptance rate which might drastically differ from the usual 0.234. Specifically, we prove that as d → ∞ the sequence of stochastic processes formed by say the i ∗ th component of each Markov chain usually converges to a Langevin diffusion process with a new speed measure υ , except in particular cases where it converges to a one-dimensional Metropolis algorithm with acceptance rule α ∗ . We also discuss the use of inhomogeneous proposals, which might prove to be essential in specific cases.