Bayesian inference has become increasingly popular as a tool to solve inverse problems, largely due to its ability to quantify the uncertainty in the solutions obtained. In many practical problems such as image reconstructions, the unknowns are often of infinite dimension, i.e., functions of space and/or time. Many existing methods developed for finite dimensional problems may become problematic in the infinite dimensional setting and thus new techniques must be developed to address such problems. In this talk we shall discuss several critical issues associated with the infinite dimensional problems and some efforts made to address them. First we introduce a family of hybrid priors for modeling functions that are subject to sharp jumps. We then present an efficient adaptive MCMC algorithm that is specifically designed for function space inference. Finally, we apply the Bayesian inference methods to a medical image reconstruction problem.

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