  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
$\mathbb{R}^N$-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.
