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We provide a brief overview of both Bayes and classical model selection. Weargue tentatively that model selection has at least two major goals, that offinding the correct model or predicting well, and that in general both thesegoals may not be achieved in an optimum manner by a single model selectionrule. We discuss, briefly but critically, through a study of well-known modelselection rules like AIC, BIC, DIC and Lasso, how these different goals arepursued in each paradigm. We introduce some new definitions of consistency,results and conjectures about consistency in high dimensional model selectionproblems. Finally we discuss some new or recent results in Full Bayes andEmpirical Bayes multiple testing, and cross-validation. We show that when thenumber of parameters tends to infinity at a smaller rate than sample size, thenit is best from the point of view of consistency to use most of the data forinference and only a negligible proportion to make an improper prior proper.

Model Selection and Multiple Testing - A Bayesian and Empirical Bayes Overview and some New Results