  The computational complexity of MCMC methods for the exploration of complex
probability measures is a challenging and important problem. A challenge of
particular importance arises in Bayesian inverse problems where the target
distribution may be supported on an infinite dimensional space. In practice
this involves the approximation of measures defined on sequences of spaces of
increasing dimension. Motivated by an elliptic inverse problem with
non-Gaussian prior, we study the design of proposal chains for the
Metropolis-Hastings algorithm with dimension independent performance.
Dimension-independent bounds on the Monte-Carlo error of MCMC sampling for
Gaussian prior measures have already been established. In this paper we provide
a simple recipe to obtain these bounds for non-Gaussian prior measures. To
illustrate the theory we consider an elliptic inverse problem arising in
groundwater flow. We explicitly construct an efficient Metropolis-Hastings
proposal based on local proposals, and we provide numerical evidence which
supports the theory.
