  We show that prediction performance for global-local shrinkage regression can
overcome two major difficulties of global shrinkage regression: (i) the amount
of relative shrinkage is monotone in the singular values of the design matrix
and (ii) the shrinkage is determined by a single tuning parameter.
Specifically, we show that the horseshoe regression, with heavy-tailed
component-specific local shrinkage parameters, in conjunction with a global
parameter providing shrinkage towards zero, alleviates both these difficulties
and consequently, results in an improved risk for prediction. Numerical
demonstrations of improved prediction over competing approaches in simulations
and in a pharmacogenomics data set confirm our theoretical findings.
