This paper studies the multiplicity-correction effect of standard Bayesian
variable-selection priors in linear regression. Our first goal is to clarify
when, and how, multiplicity correction happens automatically in Bayesian
analysis, and to distinguish this correction from the Bayesian Ockham's-razor
effect. Our second goal is to contrast empirical-Bayes and fully Bayesian
approaches to variable selection through examples, theoretical results and
simulations. Considerable differences between the two approaches are found. In
particular, we prove a theorem that characterizes a surprising aymptotic
discrepancy between fully Bayes and empirical Bayes. This discrepancy arises
from a different source than the failure to account for hyperparameter
uncertainty in the empirical-Bayes estimate. Indeed, even at the extreme, when
the empirical-Bayes estimate converges asymptotically to the true
variable-inclusion probability, the potential for a serious difference remains.Comment: Published in at http://dx.doi.org/10.1214/10-AOS792 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
