Variational Bayesian Approximation methods for inverse problems

International audienceIn this paper, first the basics of the Bayesian inference for linear inverse problems are presented. The inverse problems we consider are, for example, signal deconvolution, image restoration or image reconstruction in Computed Tomography (CT). The main point to discuss then is the prior modeling of signals and images. We consider two classes of priors: \emph{simple} or \emph{hierarchical with hidden variables}. For practical applications, we need also to consider the estimation of the hyper parameters. Finally, we see that we have to infer simultaneously the unknowns, the hidden variables and the hyper parameters. Very often, the expression of the joint posterior law of all the unknowns is too complex to be handled directly. Indeed, rarely we can obtain analytical solutions to any point estimators such the Maximum A posteriori (MAP) or Posterior Mean (PM). Three main tools can then be used: Laplace approximation (LAP), Markov Chain Monte Carlo (MCMC) and Bayesian Variational Approximations (BVA). To illustrate all these aspects, we will consider a deconvolution problem where we know that the input signal is \emph{sparse} and propose to use a Student-t distribution for that. Then, to handle the Bayesian computations with this model, we use the property of Student-t which is modelling it via an infinite mixture of Gaussians, introducing thus hidden variables which are the variances. Then, the expression of the joint posterior of the input signal samples, the hidden variables (which are here the inverse variances of those samples) and the hyper-parameters of the problem (for example the variance of the noise) is given. From this point, we will present the joint maximization by alternate optimization and the three possible approximation methods. Finally, the proposed methodology is applied in different applications such as mass spectrometry, spectrum estimation of quasi periodic biological signals and X ray computed tomography