  In a non supervised Bayesian estimation approach for inverse problems in
imaging systems, one tries to estimate jointly the unknown image pixels $f$ and
the hyperparameters $\theta$ given the observed data $g$ and a model $M$
linking these quantities. This is, in general, done through the joint posterior
law $p(f,\theta|g;M)$. 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 or integration of
multivariate probability laws. In any of these cases, we need to do
approximations. Laplace approximation and sampling by MCMC are two
approximation methods, respectively analytical and numerical, which have been
used before with success for this task. In this paper, we explore the
possibility of approximating this joint law by a separable one in $f$ and in
$\theta$. 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. To illustrate more in detail this approach, we
consider the case of image restoration by simple or myopic deconvolution with
separable, simple markovian or hidden markovian models.
