MCMC Methods for Functions: Modifying Old Algorithms to Make Them Faster

摘要: Many problems arising in applications result 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 mesh refinement dictated by description of unknown function. We describe an approach modifying whole range methods, applicable whenever target measure has density with respect Gaussian process or random field reference measure, which ensures that their speed convergence is robust refinement. Gaussian processes fields are whose marginal distributions, when evaluated at any finite set NNpoints, ℝ^N-valued Gaussians. The algorithmic we not only desired but also some useful non-Gaussian measures constructed through truncation. In interest data often sparse prior specification essential part overall modelling strategy. These Gaussian-based very flexible tool, finding wide-ranging application. shown estimation, assimilation fluid mechanics, subsurface geophysics image registration. The key design principle formulate method so it is, principle, functions; this may be achieved use proposals based on carefully chosen time-discretizations stochastic dynamical systems exactly preserve measure. Taking leads many new can implemented via minor modification existing algorithms, yet show enormous speed-up wide applied problems.