Hierarchical Bayesian formulation of Sparse Signal Recovery algorithms using scale mixture priors

In the recent past, the Sparse Signal Recovery (SSR) problem has been very well studied using penalized regression approaches with different choice of penalty functions. In this work we revisit these penalized regression formulations in a Bayesian framework with suitable choice of supergaussian prior distributions. We introduce a generalized scale mixture framework, and provide connections with well known norm minimization based SSR algorithms. Of particular interest is the re-weighted ℓ₁ approach. The scale mixture representation allows us to formulate the corresponding Type II version of these algorithms, following the hierarchical bayesian framework of Sparse Bayesian Learning (SBL) and enable a comparison of Type I versus Type II approaches.