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Optimal scaling of random walk Metropolis algorithms with discontinuous target densities

Neal, Peter and Roberts, Gareth and Yuen, Wai Kong (2012) Optimal scaling of random walk Metropolis algorithms with discontinuous target densities. Annals of Applied Probability, 22 (5). pp. 1880-1927. ISSN 1050-5164

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Abstract

We consider the optimal scaling problem for high-dimensional random walk Metropolis (RWM) algorithms where the target distribution has a discontinuous probability density function. Almost all previous analysis has focused upon continuous target densities. The main result is a weak convergence result as the dimensionality d of the target densities converges to ∞. In particular, when the proposal variance is scaled by d−2, the sequence of stochastic processes formed by the first component of each Markov chain converges to an appropriate Langevin diffusion process. Therefore optimizing the efficiency of the RWM algorithm is equivalent to maximizing the speed of the limiting diffusion. This leads to an asymptotic optimal acceptance rate of e−2 (=0.1353) under quite general conditions. The results have major practical implications for the implementation of RWM algorithms by highlighting the detrimental effect of choosing RWM algorithms over Metropolis-within-Gibbs algorithms.

Item Type: Article
Journal or Publication Title: Annals of Applied Probability
Uncontrolled Keywords: Random walk Metropolis ; Markov chain Monte Carlo ; optimal scaling
Subjects: Q Science > QA Mathematics
Departments: Faculty of Science and Technology > Mathematics and Statistics
ID Code: 59734
Deposited By: ep_importer_pure
Deposited On: 31 Oct 2012 11:15
Refereed?: Yes
Published?: Published
Last Modified: 09 Oct 2013 15:43
Identification Number:
URI: http://eprints.lancs.ac.uk/id/eprint/59734

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