Many Bayesian inference problems require exploring the posterior distribution of high-dimensional parameters,  which in principle can be described as functions.  Formulating algorithms which are defined on function space yields dimension-independent algorithms. 
 
By exploiting the intrinsic low dimensionality of the likelihood function, we introduce a newly developed suite of proposals for the Metropolis Hastings MCMC algorithm that can adapt to the complex structure of the posterior distribution, yet are defined on function space.  I will present numerical examples indicating the efficiency of these dimension-independent likelihood-informed samplers.
I will also present some applications of function-space samplers to problems relevant to numerical weather prediction and subsurface reconstruction.