  Uncovering genuine relationships between a response variable of interest and
a large collection of covariates is a fundamental and practically important
problem. In the context of Gaussian linear models, both the Bayesian and
non-Bayesian literature is well-developed and there are no substantial
differences in the model selection consistency results available from the two
schools. For the more challenging generalized linear models (GLMs), however,
Bayesian model selection consistency results are lacking in several ways. In
this paper, we construct a Bayesian posterior distribution using an appropriate
data-dependent prior and develop its asymptotic concentration properties using
new theoretical techniques. In particular, we leverage Spokoiny's powerful
non-asymptotic theory to obtain sharp quadratic approximations of the GLM's
log-likelihood function, which leads to tight bounds on the errors associated
with the model-specific maximum likelihood estimators and the Laplace
approximation of our Bayesian marginal likelihood. In turn, these improved
bounds lead to significantly stronger, near-optimal Bayesian model selection
consistency results, e.g., far weaker beta-min conditions, compared to those
available in the existing literature. In particular, our results are applicable
to the Poisson regression model, in which the score function is not
sub-Gaussian.
