  Bayesian variable selection has gained much empirical success recently in a
variety of applications when the number $K$ of explanatory variables
$(x_1,...,x_K)$ is possibly much larger than the sample size $n$. For
generalized linear models, if most of the $x_j$'s have very small effects on
the response $y$, we show that it is possible to use Bayesian variable
selection to reduce overfitting caused by the curse of dimensionality $K\gg n$.
In this approach a suitable prior can be used to choose a few out of the many
$x_j$'s to model $y$, so that the posterior will propose probability densities
$p$ that are ``often close'' to the true density $p^*$ in some sense. The
closeness can be described by a Hellinger distance between $p$ and $p^*$ that
scales at a power very close to $n^{-1/2}$, which is the ``finite-dimensional
rate'' corresponding to a low-dimensional situation. These findings extend some
recent work of Jiang [Technical Report 05-02 (2005) Dept. Statistics,
Northwestern Univ.] on consistency of Bayesian variable selection for binary
classification.
