  We present a sparse estimation and dictionary learning framework for
compressed fiber sensing based on a probabilistic hierarchical sparse model. To
handle severe dictionary coherence, selective shrinkage is achieved using a
Weibull prior, which can be related to non-convex optimization with $p$-norm
constraints for $0 < p < 1$. In addition, we leverage the specific dictionary
structure to promote collective shrinkage based on a local similarity model.
This is incorporated in form of a kernel function in the joint prior density of
the sparse coefficients, thereby establishing a Markov random field-relation.
Approximate inference is accomplished using a hybrid technique that combines
Hamilton Monte Carlo and Gibbs sampling. To estimate the dictionary parameter,
we pursue two strategies, relying on either a deterministic or a probabilistic
model for the dictionary parameter. In the first strategy, the parameter is
estimated based on alternating estimation. In the second strategy, it is
jointly estimated along with the sparse coefficients. The performance is
evaluated in comparison to an existing method in various scenarios using
simulations and experimental data.
