  In this paper we address the problem of the prohibitively large computational
cost of existing Markov chain Monte Carlo methods for large--scale applications
with high dimensional parameter spaces, e.g. in uncertainty quantification in
porous media flow. We propose a new multilevel Metropolis-Hastings algorithm,
and give an abstract, problem dependent theorem on the cost of the new
multilevel estimator based on a set of simple, verifiable assumptions. For a
typical model problem in subsurface flow, we then provide a detailed analysis
of these assumptions and show significant gains over the standard
Metropolis-Hastings estimator. Numerical experiments confirm the analysis and
demonstrate the effectiveness of the method with consistent reductions of more
than an order of magnitude in the cost of the multilevel estimator over the
standard Metropolis-Hastings algorithm for tolerances $\varepsilon < 10^{-2}$.
