  We consider Bayesian variable selection in sparse high-dimensional
regression, where the number of covariates $p$ may be large relative to the
samples size $n$, but at most a moderate number $q$ of covariates are active.
Specifically, we treat generalized linear models. For a single fixed sparse
model with well-behaved prior distribution, classical theory proves that the
Laplace approximation to the marginal likelihood of the model is accurate for
sufficiently large sample size $n$. We extend this theory by giving results on
uniform accuracy of the Laplace approximation across all models in a
high-dimensional scenario in which $p$ and $q$, and thus also the number of
considered models, may increase with $n$. Moreover, we show how this connection
between marginal likelihood and Laplace approximation can be used to obtain
consistency results for Bayesian approaches to variable selection in
high-dimensional regression.
