  Statistical inference based on moment conditions and estimating equations is
of substantial interest when it is difficult to specify a full probabilistic
model. We propose a Bayesian flavored model selection framework based on
(quasi-)posterior probabilities from the Bayesian Generalized Method of Moments
(BGMM), which allows us to incorporate two important advantages of a Bayesian
approach: the expressiveness of posterior distributions and the convenient
computational method of Markov Chain Monte Carlo (MCMC). Theoretically we show
that BGMM can achieve the posterior consistency for selecting the unknown true
model, and that it possesses a Bayesian version of the oracle property, i.e.
the posterior distribution for the parameter of interest is asymptotically
normal and is as informative as if the true model were known. In addition, we
show that the proposed quasi-posterior is valid to be interpreted as an
approximate posterior distribution given a data summary. Our applications
include modeling of correlated data, quantile regression, and graphical models
based on partial correlations. We demonstrate the implementation of the BGMM
model selection through numerical examples.
