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

Sherlock, Chris and Roberts, Gareth (2009) Optimal scaling of the random walk Metropolis on unimodal elliptically symmetric targets. Bernoulli, 15 (3). pp. 774-798. ISSN 1350-7265

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    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 which 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.

    Item Type: Article
    Journal or Publication Title: Bernoulli
    Uncontrolled Keywords: Random walk Metropolis ; optimal scaling ; optimal acceptance rate.
    Subjects: Q Science > QA Mathematics
    Departments: Faculty of Science and Technology > Mathematics and Statistics
    ID Code: 20931
    Deposited By: Dr Chris Sherlock
    Deposited On: 04 Dec 2008 09:38
    Refereed?: Yes
    Published?: Published
    Last Modified: 09 Oct 2013 15:41
    Identification Number:
    URI: http://eprints.lancs.ac.uk/id/eprint/20931

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