Bayesian Likelihoods for Moment Condition Models

Bayesian inference in moment condition models is difficult to implement. For these models, a posterior distribution cannot be calculated because the likelihood function has not been fully specified. In this paper, we obtain a class of likelihoods by formal Bayesian calculations that take into account the semiparametric nature of the problem. The likelihoods are derived by integrating out the nuisance parameters with respect to a maximum entropy tilted prior on the space of distribution. The result is a unification that uncovers a mapping between priors and likelihood functions. We show that there exist priors such that the likelihoods are closely connected to Generalized Empirical Likelihood (GEL) methods.