  The SLOPE estimates regression coefficients by minimizing a regularized
residual sum of squares using a sorted-$\ell_1$-norm penalty. The SLOPE
combines testing and estimation in regression problems. It exhibits suitable
variable selection and prediction properties, as well as minimax optimality.
This paper introduces the Bayesian SLOPE procedure for linear regression. The
classical SLOPE estimate is the posterior mode in the normal regression problem
with an appropriate prior on the coefficients. The Bayesian SLOPE considers the
full Bayesian model and has the advantage of offering credible sets and
standard error estimates for the parameters. Moreover, the hierarchical
Bayesian framework allows for full Bayesian and empirical Bayes treatment of
the penalty coefficients; whereas it is not clear how to choose these
coefficients when using the SLOPE on a general design matrix. A direct
characterization of the posterior is provided which suggests a Gibbs sampler
that does not involve latent variables. An efficient hybrid Gibbs sampler for
the Bayesian SLOPE is introduced. Point estimation using the posterior mean is
highlighted, which automatically facilitates the Bayesian prediction of future
observations. These are demonstrated on real and synthetic data.
