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Bayes and empirical-Bayes multiplicity adjustment in the variable-selection problem. (English) Zbl 1200.62020
Summary: 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.

62F15 Bayesian inference
62C12 Empirical decision procedures; empirical Bayes procedures
62J05 Linear regression; mixed models
62J15 Paired and multiple comparisons; multiple testing
65C60 Computational problems in statistics (MSC2010)
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