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