  Linear models with a growing number of parameters have been widely used in
modern statistics. One important problem about this kind of model is the
variable selection issue. Bayesian approaches, which provide a stochastic
search of informative variables, have gained popularity. In this paper, we will
study the asymptotic properties related to Bayesian model selection when the
model dimension $p$ is growing with the sample size $n$. We consider $p\le n$
and provide sufficient conditions under which: (1) with large probability, the
posterior probability of the true model (from which samples are drawn)
uniformly dominates the posterior probability of any incorrect models; and (2)
with large probability, the posterior probability of the true model converges
to one. Both (1) and (2) guarantee that the true model will be selected under a
Bayesian framework. We also demonstrate several situations when (1) holds but
(2) fails, which illustrates the difference between these two properties.
Simulated examples are provided to illustrate the main results.
