  This paper presents a study of the large-sample behavior of the posterior
distribution of a structural parameter which is partially identified by moment
inequalities. The posterior density is derived based on the limited information
likelihood. The posterior distribution converges to zero exponentially fast on
any $\delta$-contraction outside the identified region. Inside, it is bounded
below by a positive constant if the identified region is assumed to have a
nonempty interior. Our simulation evidence indicates that the Bayesian approach
has advantages over frequentist methods, in the sense that, with a proper
choice of the prior, the posterior provides more information about the true
parameter inside the identified region. We also address the problem of moment
and model selection. Our optimality criterion is the maximum posterior
procedure and we show that, asymptotically, it selects the true moment/model
combination with the most moment inequalities and the simplest model.
