We study linear models under heavy-tailed priors from a probabilistic
viewpoint. Instead of computing a single sparse most probable (MAP) solution as
in standard compressed sensing, the focus in the Bayesian framework shifts
towards capturing the full posterior distribution on the latent variables,
which allows quantifying the estimation uncertainty and learning model
parameters using maximum likelihood. The exact posterior distribution under the
sparse linear model is intractable and we concentrate on a number of
alternative variational Bayesian techniques to approximate it.