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

TL;DR

This paper develops a new approach to understanding the optimal scaling of the random walk Metropolis algorithm for elliptically symmetric unimodal targets, confirming the 0.234 acceptance rate rule in many cases without relying on diffusion limits.

Contribution

It introduces explicit formulae for efficiency and acceptance rates, and characterizes when the 0.234 rule holds or fails for a broad class of target densities.

Findings

The 0.234 acceptance rate rule is verified for elliptically symmetric targets.

Explicit efficiency formulas are derived as functions of scaling.

Conditions where the 0.234 rule does not hold are characterized.

Abstract

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.