  We study high-dimensional Bayesian linear regression with a general beta
prime distribution for the scale parameter. Under the assumption of sparsity,
we show that appropriate selection of the hyperparameters in the beta prime
prior leads to the (near) minimax posterior contraction rate when $p \gg n$.
For finite samples, we propose a data-adaptive method for estimating the
hyperparameters based on marginal maximum likelihood (MML). This enables our
prior to adapt to both sparse and dense settings, and under our proposed
empirical Bayes procedure, the MML estimates are never at risk of collapsing to
zero. We derive efficient Monte Carlo EM and variational EM algorithms for
implementing our model, which are available in the R package NormalBetaPrime.
Simulations and analysis of a gene expression data set illustrate our model's
self-adaptivity to varying levels of sparsity and signal strengths.
