**11. Conclusions**

Mainly, in this chapter, first we described inverse problems and gave a few classical examples such as deconvolution, image restoration, computed tomography X-ray image reconstruction, Fourier synthesis inversion problem which arise in many imaging systems. Then, we mentioned that there are two classes of methods for inverse problems: deterministic regularization and Bayesian inference methods. Then, we started by describing the Bayesian parameter estimation. The main parts of the chapter is focused on Bayesian inference for inverse problems. We saw that the main difficulty is the great dimension of unknown quantities and the appropriate choice of the prior law. For this, first we described many simple and hierarchical prior models which are used in real applications. For the second main difficulty, which is the computational aspects, we described different approximate Bayesian computations

(ABC) and in particular the variational Bayesian approximation (VBA) methods and showed how to use these methods, for example for hyperparameter estimation or for large scale inverse problems.
