  Most estimates for penalised linear regression can be viewed as posterior
modes for an appropriate choice of prior distribution. Bayesian shrinkage
methods, particularly the horseshoe estimator, have recently attracted a great
deal of attention in the problem of estimating sparse, high-dimensional linear
models. This paper extends these ideas, and presents a Bayesian grouped model
with continuous global-local shrinkage priors to handle complex group
hierarchies that include overlapping and multilevel group structures. As the
posterior mean is never a sparse estimate of the linear model coefficients, we
extend the recently proposed decoupled shrinkage and selection (DSS) technique
to the problem of selecting groups of variables from posterior samples. To
choose a final, sparse model, we also adapt generalised information criteria
approaches to the DSS framework. To ensure that sparse groups, in which only a
few predictors are active, can be effectively identified, we provide an
alternative degrees of freedom estimator for sparse Bayesian linear models that
takes into account the effects of shrinkage on the model coefficients.
Simulations and real data analysis using our proposed method show promising
performance in terms of correct identification of active and inactive groups,
and prediction, in comparison with a Bayesian grouped slab-and-spike approach.
