Abstract

The random walk Metropolis (RWM) is one of the most commonly used Metropolis-Hastings algorithms, and choosing the appropriate scaling for the proposal is an important practical problem. Previous theoretical approaches have focussed on high-dimensional algorithms and have revolved around a diffusion approximation of the trajectory. For certain specific classes of targets it has been possible to show that the algorithm is optimal when the acceptance rate is approximately 0.234.

We develop a novel approach which avoids the need for diffusion limits. Focussing on spherically symmetric targets, it is possible to derive simple exact formulae for efficiency and acceptance rate for a “real” RWM algorithm, as opposed to a limit process. The limiting behaviour of these formulae can then be explored. This in some sense “simpler” approach allows important general intuitions as to when and why the 0.234 rule holds, when the rule fails, and what may happen when it does fail. By extending the theory to include elliptically symmetric targets we obtain further intuitions about the role of the proposal’s shape.