  Bayesian variable selection methods are powerful techniques for fitting and
inferring on sparse high-dimensional linear regression models. However, many
are computationally intensive or require restrictive prior distributions on
model parameters. In this paper, we proposed a computationally efficient and
powerful Bayesian approach for sparse high-dimensional linear regression.
Minimal prior assumptions on the parameters are required through the use of
plug-in empirical Bayes estimates of hyperparameters. Efficient maximum a
posteriori (MAP) estimation is completed through a Parameter-Expanded
Expectation-Conditional-Maximization (PX-ECM) algorithm. The PX-ECM results in
a robust computationally efficient coordinate-wise optimization which -- when
updating the coefficient for a particular predictor -- adjusts for the impact
of other predictor variables. The completion of the E-step uses an approach
motivated by the popular two-group approach to multiple testing. The result is
a PaRtitiOned empirical Bayes Ecm (PROBE) algorithm applied to sparse
high-dimensional linear regression, which can be completed using one-at-a-time
or all-at-once type optimization. We compare the empirical properties of PROBE
to comparable approaches with numerous simulation studies and analyses of
cancer cell drug responses. The proposed approach is implemented in the R
package probe.
