Title: Bayesian model selection consistency and oracle inequality with intractable marginal likelihood,
Abstract: In this article, we investigate large sample properties of model selection
procedures in a general Bayesian framework when a closed form expression of the
marginal likelihood function is not available or a local asymptotic quadratic
approximation of the log-likelihood function does not exist. Under appropriate
identifiability assumptions on the true model, we provide sufficient conditions
for a Bayesian model selection procedure to be consistent and obey the Occam's
razor phenomenon, i.e., the probability of selecting the "smallest" model that
contains the truth tends to one as the sample size goes to infinity. In order
to show that a Bayesian model selection procedure selects the smallest model
containing the truth, we impose a prior anti-concentration condition, requiring
the prior mass assigned by large models to a neighborhood of the truth to be
sufficiently small. In a more general setting where the strong model
identifiability assumption may not hold, we introduce the notion of local
Bayesian complexity and develop oracle inequalities for Bayesian model
selection procedures. Our Bayesian oracle inequality characterizes a trade-off
between the approximation error and a Bayesian characterization of the local
complexity of the model, illustrating the adaptive nature of averaging-based
Bayesian procedures towards achieving an optimal rate of posterior convergence.
Specific applications of the model selection theory are discussed in the
context of high-dimensional nonparametric regression and density regression
where the regression function or the conditional density is assumed to depend
on a fixed subset of predictors. As a result of independent interest, we
propose a general technique for obtaining upper bounds of certain small ball
probability of stationary Gaussian processes.