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.

Bayesian penalized regression techniques, such as the Bayesian lasso and the Bayesian horseshoe estimator, have recently received a significant amount of attention in the statistics literature. However, software implementing state-of-the-art Bayesian penalized regression, outside of general purpose Markov chain Monte Carlo platforms such as STAN, is relatively rare. This paper introduces bayesreg, a new toolbox for fitting Bayesian penalized regression models with continuous shrinkage prior densities. The toolbox features Bayesian linear regression with Gaussian or heavy-tailed error models and Bayesian logistic regression with ridge, lasso, horseshoe and horseshoe$+$ estimators. The toolbox is free, open-source and available for use with the MATLAB and R numerical platforms.