Model Selection in the Sparsity Context for Inverse Problems in Bayesian Framework

Abstract

The Bayesian approach is considered for inverse problems with a typical forward model accounting for errors and a priori sparse solutions. Solutions with sparse structure are enforced using heavy-tailed prior distributions. The particular case of such prior expressed via normal variance mixtures with conjugate laws for the mixing distribution is the main interest of this paper. Such a prior is considered in this paper, namely, the Student-t distribution. Iterative algorithms are derived via posterior mean estimation. The mixing distribution parameters appear in updating equations and are also used for the initialization. For the choice of mixing distribution parameters, three model selection strategies are considered: (i) parameters approximating the mixing distribution with Jeffrey law, i.e., keeping the mixing distribution well defined but as close as possible to the Jeffreys priors, (ii) based on the prior distribution form, fixing the parameters corresponding to the form inducing the most sparse solution and (iii) based on the sparsity mechanism, fixing the hyperparameters using the statistical measures of the mixing and prior distribution. For each strategy of model selection, the theoretical advantages and drawbacks are discussed and the corresponding simulations are reported for a 1D direct sparsity application in a biomedical context. We show that the third strategy seems to provide the best parameter selection strategy for this context.