Abstract:  In this talk, I will start with a review of the characteristics of classical inverse problems before moving into Bayesian methods for inverse problems. The first Bayesian methodology I will discuss is the use of Gaussian Markov random fields (GMRFs) for modeling the prior. Then, with the prior in hand, I will proceed to the main part of the talk, which focuses on the problem of sampling from the posterior density function (or simply the posterior). I will assume that both the measurement error and prior variances are unknown and place hyper-prior probability density functions on these scalar parameters. The resulting hierarchical Bayesian posterior is non-Gaussian, but lends itself well to the use of the Gibbs sampler. I will present this Gibbs sampler and then discuss its drawbacks in terms of algorithmic performance. I will then present two alternative MCMC methods, both of which make use of marginalization, and which have better performance characteristics than the Gibbs sampler.