Model Selection for Likelihood-free Bayesian Methods Based on Moment Conditions: Theory and Numerical Examples

An important practice in statistics is to use robust likelihood-free methods, such as the estimating equations, which only require assumptions on the moments instead of specifying the full probabilistic model. We propose a Bayesian avored model selection approach for such likelihood-free methods, based on (quasi-)posterior probabilities from the Bayesian Generalized Method of Moments (BGMM). This novel concept allows us to incorporate two important advantages of a Bayesian approach: the expressiveness of posterior distributions and the convenient computational method of MCMC. Many different applications are possible, including modeling the correlated longitudinal data, the quantile regression, and the graphical models based on partial correlation. We demonstrate numerically how our method works in these applications. Under mild conditions, we show that theoretically the 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 conditional distribution given a data summary.