Abstract
In this chapter, we address the challenge of exploring the posterior distributions of Bayesian inverse problems with computationally intensive forward models. As the central example, we explore a posterior distribution over a 2-d lattice resulting from an application in electrical impedance tomography (EIT). We first describe a basic single site Metropolis scheme for sampling the posterior distribution, essentially using the approach detailed in the original Metropolis et al. paper of 1953. We then go on to explore alternative, reversible updating schemes for MCMC. This includes multivariate updating, delayed acceptance, and schemes that leverage a crude, coarsened forward model for the EIT application. In particular, the adaptive, multiple-step, delayed acceptance scheme shows substantial speed up relative to other approaches.