Classical methods for inverse problems are mainly based on regularization theory. In particular those which are based on optimization of a criterion with two parts: a data-model matching and a regularization term. Different choices for these two terms and great number of optimization algorithms have been proposed. When these two terms are distance or divergence measures, they can have a Bayesian Maximum A Posteriori (MAP) interpretation where these two terms correspond, respectively, to the likelihood and prior probability models.

In many Bayesian computations, first, we obtain the expression of the joint distribution of all the unknown variables given the observed data. In general, this expression is not separable in those variables. Thus, obtaining their marginals for each variable and computing the expectations are difficult and costly. This problem becomes even more difficult in high dimensional quandaries, which is an important issue in inverse problems. We may then try to propose a surrogate expression with which we can do approximate computations. Often a separable expression approximation can be useful enough. The Variational Bayesian Approximation (VBA) is a technique that approximates the joint distribution $p$ with an easier, for example separable, one $q$ by minimizing Kullback–Leibler Divergence $KL(q|p)$. When $q$ is separable in all the variables, the approximation is also called Mean Field Approximation (MFA) and so $q$ is the product of the approximated marginals. A first standard and general algorithm is alternate optimization of $KL(q|p)$ with respect to $q_i$. A second general approach is its optimization in the Riemannian manifold. However, in this paper, for practical reasons, we consider the case where $p$ is in the exponential family and so is $q$. For this case, $KL(q|p)$ becomes a function of the parameters $\thetab$ of the exponential family. Then, we can use any other optimization algorithm to obtain those parameters. In this paper, we compare three optimization algorithms: standard alternate optimization, a gradient-based algorithm and a natural gradient algorithm and study their relative performances on three examples.