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
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.
Citation
S. L. Cotter. G. O. Roberts. A. M. Stuart. D. White. "MCMC Methods for Functions: Modifying Old Algorithms to Make Them Faster." Statist. Sci. 28 (3) 424 - 446, August 2013. https://doi.org/10.1214/13-STS421
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