  Maximum a posteriori (MAP) estimation, like all Bayesian methods, depends on
prior assumptions. These assumptions are often chosen to promote specific
features in the recovered estimate. The form of the chosen prior determines the
shape of the posterior distribution, thus the behavior of the estimator and
complexity of the associated optimization problem. Here, we consider a family
of Gaussian hierarchical models with generalized gamma hyperpriors designed to
promote sparsity in linear inverse problems. By varying the hyperparameters, we
move continuously between priors that act as smoothed $\ell_p$ penalties with
flexible $p$, smoothing, and scale. We then introduce a predictor-corrector
method that tracks MAP solution paths as the hyperparameters vary. Path
following allows a user to explore the space of possible MAP solutions and to
test the sensitivity of solutions to changes in the prior assumptions. By
tracing paths from a convex region to a non-convex region, the user can find
local minimizers in strongly sparsity promoting regimes that are consistent
with a convex relaxation derived using related prior assumptions. We show
experimentally that these solutions. are less error prone than direct
optimization of the non-convex problem.
