  In a non supervised Bayesian estimation approach for inverse problems in
imaging systems, one tries to estimate jointly the unknown image pixels $\fb$
and the hyperparameters $\thetab$. This is, in general, done through the joint
posterior law $p(\fb,\thetab|\gb)$. The expression of this joint law is often
very complex and its exploration through sampling and computation of the point
estimators such as MAP and posterior means need either optimization of non
convex criteria or int\'egration of non Gaussian and multi variate probability
laws. In any of these cases, we need to do approximations. We had explored
before the possibilities of Laplace approximation and sampling by MCMC. In this
paper, we explore the possibility of approximating this joint law by a
separable one in $\fb$ and in $\thetab$. This gives the possibility of
developing iterative algorithms with more reasonable computational cost, in
particular, if the approximating laws are choosed in the exponential conjugate
families. The main objective of this paper is to give details of different
algorithms we obtain with different choices of these families.
