A Bayesian approach to characterizing uncertainty in inverse problems using coarse and fine-scale information

The Bayesian approach allows one to easily quantify uncertainty, at least in theory. In practice, however, MCMC can be computationally expensive, particularly in complicated inverse problems. Here we present methodology for improving the speed and efficiency of an MCMC analysis by combining runs on different scales. By using a coarser scale, the chain can run faster (particularly when there is an external forward simulator involved in the likelihood evaluation) and better explore the posterior, being less likely to become stuck in local maxima. We discuss methods for linking the coarse chain back to the original fine scale chain of interest. The resulting coupled chain can thus be run more efficiently without sacrificing the accuracy achieved at the finer scale.