A full Bayesian approach for inverse problems

The main object of this paper is to present some general concepts of Bayesian inference and more specifically the estimation of the hyperparameters in inverse problems. We consider a general linear situation where we are given some data $\yb$ related to the unknown parameters $\xb$ by $\yb=\Ab \xb+\nb$ and where we can assign the probability laws $p(\xb|\thetab)$, $p(\yb|\xb,\betab)$, $p(\betab)$ and $p(\thetab)$. The main discussion is then how to infer $\xb$, $\thetab$ and $\betab$ either individually or any combinations of them. Different situations are considered and discussed. As an important example, we consider the case where $\theta$ and $\beta$ are the precision parameters of the Gaussian laws to whom we assign Gamma priors and we propose some new and practical algorithms to estimate them simultaneously. Comparisons and links with other classical methods such as maximum likelihood are presented. Keywords: Bayesian inference, Hyperparameter estimation, Inverse problems, Maximum likelihood.