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