Bayesian Inference for Inverse Problems

Inverse problems arise everywhere we have indirect measurement. Regularization and Bayesian inference methods are two main approaches to handle inverse problems. Bayesian inference approach is more general and has much more tools for developing efficient methods for difficult problems. In this chapter, first, an overview of the Bayesian parameter estimation is presented, then we see the extension for inverse problems. The main difficulty is the great dimension of unknown quantity and the appropriate choice of the prior law. The second main difficulty is the computational aspects. Different approximate Bayesian computations and in particular the variational Bayesian approximation (VBA) methods are explained in details.