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.

本文研究标准 Bayesian 变量选择先验在线性回归中的多重校正效应，澄清了 Bayesian 方法中的多重校正及 Ockham's-razor 的区别，同时通过实例、理论和模拟对比了经验 Bayes 和全 Bayes 方法。