Bayesian Computation and the Linear Model

This paper is a review of computational strategies for Bayesian shrinkage and variable selection in the linear model. Our focus is less on traditional MCMC methods, which are covered in depth by earlier review papers. Instead, we focus more on recent innovations in stochastic search and adaptive MCMC, along with some comparatively new research on shrinkage priors. One of our conclusions is that true MCMC seems inferior to stochastic search if one’s goal is to discover good models, but that stochastic search can result in biased estimates of variable inclusion probabilities. We also find reasons to question the accuracy of inclusion probabilities generated by traditional MCMC on high-dimensional, nonorthogonal problems, though the matter is far from settled. Some key words: adaptive MCMC; linear models; shrinkage priors; stochastic search; variable selection