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MCMC methods for functions : modifying old algorithms to make them faster
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Cotter, Simon L., Roberts, Gareth O., Stuart, A. M. and White, David (2013) MCMC methods for functions : modifying old algorithms to make them faster. Statistical Science, Volume 28 (Number 3). pp. 424-446. doi:10.1214/13-STS421 ISSN 0883-4237.
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Official URL: http://dx.doi.org/10.1214/13-STS421
Abstract
Many problems arising in applications result in the need to probe a probability distribution for functions. Examples include Bayesian nonparametric statistics and conditioned diffusion processes. Standard MCMC algorithms typically become arbitrarily slow under the mesh refinement dictated by nonparametric description of the unknown function. We describe an approach to modifying a whole range of MCMC methods, applicable whenever the target measure has density with respect to a Gaussian process or Gaussian random field reference measure, which ensures that their speed of convergence is robust under mesh refinement.
Gaussian processes or random fields are fields whose marginal distributions, when evaluated at any finite set of N points, are R N -valued Gaussians. The algorithmic approach that we describe is applicable not only when the desired probability measure has density with respect to a Gaussian process or Gaussian random field reference measure, but also to some useful non-Gaussian reference measures constructed through random truncation. In the applications of interest the data is often sparse and the prior specification is an essential part of the overall modelling strategy. These Gaussian-based reference measures are a very flexible modelling tool, finding wide-ranging application. Examples are shown in density estimation, data assimilation in fluid mechanics, subsurface geophysics and image registration.
The key design principle is to formulate the MCMC method so that it is, in principle, applicable for functions; this may be achieved by use of proposals based on carefully chosen time-discretizations of stochastic dynamical systems which exactly preserve the Gaussian reference measure. Taking this approach leads to many new algorithms which can be implemented via minor modification of existing algorithms, yet which show enormous speed-up on a wide range of applied problems.
Item Type: | Journal Article | ||||
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Divisions: | Faculty of Science, Engineering and Medicine > Science > Statistics | ||||
Journal or Publication Title: | Statistical Science | ||||
Publisher: | American Institute of Mathematical Statistics | ||||
ISSN: | 0883-4237 | ||||
Official Date: | 28 August 2013 | ||||
Dates: |
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Volume: | Volume 28 | ||||
Number: | Number 3 | ||||
Page Range: | pp. 424-446 | ||||
DOI: | 10.1214/13-STS421 | ||||
Status: | Peer Reviewed | ||||
Publication Status: | Published | ||||
Access rights to Published version: | Restricted or Subscription Access |
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