Optimal scaling of the random walk Metropolis on unimodal elliptically symmetric targets.
By Chris Sherlock, Gareth O. Roberts. 2009
58 citations recorded by the source corpus.
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.
Subfields: Condensed Matter Physics, Mathematical Physics, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/dd7bbdc9b0cb716b6f3b4a3b565cd6ed555708cc
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In Mathematical Physics
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Citations in the corpus
- A tutorial on adaptive MCMC (2008, background): “Such a simple criterion, which is at least completely supported by theory in some scenarios (Roberts et al. 1997; Sherlock and Roberts 2006; Roberts and Rosenthal 1998; Bédard 2006) and proves to be more universal in practice, is the expected acceptance probability of the MH algorithm for random…”