  Scale-mixture shrinkage priors have recently been shown to possess robust
empirical performance and excellent theoretical properties such as model
selection consistency and (near) minimax posterior contraction rates. In this
paper, the normal-compound gamma prior (NCG) resulting from compounding on the
respective inverse-scale parameters with gamma distribution is used as a prior
for the scale parameter. Attractiveness of this model becomes apparent due to
its relationship to various useful models. The tuning of the hyperparameters
gives the same shrinkage properties exhibited by some other models. Using
different sets of conditions, the posterior is shown to be both strongly
consistent and have nearly-optimal contraction rates depending on the set of
assumptions. Furthermore, the Monte Carlo Markov Chain (MCMC) and Variational
Bayes algorithms are derived, then a method is proposed for updating the
hyperparameters and is incorporated into the MCMC and Variational Bayes
algorithms. Finally, empirical evidence of the attractiveness of this model is
demonstrated using both real and simulated data, to compare the predicted
results with previous models.
