Constrained empirical Bayes priors on regression coefficients
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In the context of model uncertainty and selection, empirical Bayes procedures can have undesirable properties such as extreme estimates of inclusion probabilities (Scott and Berger, 2010) or inconsistency under the null model (Liang et al., 2008). To avoid these issues, we define empirical Bayes priors with constraints that ensure that the estimates of the hyperparameters are at least as "vague" as those of proper default priors. In our examples, we observe that constrained EB procedures are better behaved than their unconstrained counterparts and that the Bayesian Information Criterion (BIC) is similar to an intuitively appealing constrained EB procedure.
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