  Motivated by the increasing use of and rapid changes in array technologies,
we consider the prediction problem of fitting a linear regression relating a
continuous outcome $Y$ to a large number of covariates $\mathbf {X}$, for
example, measurements from current, state-of-the-art technology. For most of
the samples, only the outcome $Y$ and surrogate covariates, $\mathbf {W}$, are
available. These surrogates may be data from prior studies using older
technologies. Owing to the dimension of the problem and the large fraction of
missing information, a critical issue is appropriate shrinkage of model
parameters for an optimal bias-variance trade-off. We discuss a variety of
fully Bayesian and Empirical Bayes algorithms which account for uncertainty in
the missing data and adaptively shrink parameter estimates for superior
prediction. These methods are evaluated via a comprehensive simulation study.
In addition, we apply our methods to a lung cancer data set, predicting
survival time ($Y$) using qRT-PCR ($\mathbf {X}$) and microarray ($\mathbf
{W}$) measurements.
