  As an alternative to variable selection or shrinkage in high dimensional
regression, we propose to randomly compress the predictors prior to analysis.
This dramatically reduces storage and computational bottlenecks, performing
well when the predictors can be projected to a low dimensional linear subspace
with minimal loss of information about the response. As opposed to existing
Bayesian dimensionality reduction approaches, the exact posterior distribution
conditional on the compressed data is available analytically, speeding up
computation by many orders of magnitude while also bypassing robustness issues
due to convergence and mixing problems with MCMC. Model averaging is used to
reduce sensitivity to the random projection matrix, while accommodating
uncertainty in the subspace dimension. Strong theoretical support is provided
for the approach by showing near parametric convergence rates for the
predictive density in the large p small n asymptotic paradigm. Practical
performance relative to competitors is illustrated in simulations and real data
applications.
