  In many regression settings the unknown coefficients may have some known
structure, for instance they may be ordered in space or correspond to a
vectorized matrix or tensor. At the same time, the unknown coefficients may be
sparse, with many nearly or exactly equal to zero. However, many commonly used
priors and corresponding penalties for coefficients do not encourage
simultaneously structured and sparse estimates. In this paper we develop
structured shrinkage priors that generalize multivariate normal, Laplace,
exponential power and normal-gamma priors. These priors allow the regression
coefficients to be correlated a priori without sacrificing elementwise sparsity
or shrinkage. The primary challenges in working with these structured shrinkage
priors are computational, as the corresponding penalties are intractable
integrals and the full conditional distributions that are needed to approximate
the posterior mode or simulate from the posterior distribution may be
non-standard. We overcome these issues using a flexible elliptical slice
sampling procedure, and demonstrate that these priors can be used to introduce
structure while preserving sparsity.
