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Scaling of proposals for Metropolis algorithms is an important practicalproblem in MCMC implementation. Criteria for scaling based on empiricalacceptance rates of algorithms have been found to work consistently well acrossa broad range of problems. Essentially, proposal jump sizes are increased whenacceptance rates are high and decreased when rates are low. In recent years,considerable theoretical support has been given for rules of this type whichwork on the basis that acceptance rates around 0.234 should be preferred. Thishas been based on asymptotic results that approximate high dimensionalalgorithm trajectories by diffusions. In this paper, we develop a novelapproach to understanding 0.234 which avoids the need for diffusion limits. Wederive explicit formulae for algorithm efficiency and acceptance rates asfunctions of the scaling parameter. We apply these to the family ofelliptically symmetric target densities, where further illuminating explicitresults are possible. Under suitable conditions, we verify the 0.234 rule for anew class of target densities. Moreover, we can characterise cases where 0.234fails to hold, either because the target density is too diffuse in a sense wemake precise, or because the eccentricity of the target density is too severe,again in a sense we make precise. We provide numerical verifications of ourresults.

Optimal scaling of the random walk Metropolis on elliptically symmetric unimodal targets