Bayesian and Frequentist Inference in Partially Identified Models

A large-sample approximation of the posterior distribution of partially identified structural parameters is derived for models that can be indexed by an identifiable finite-dimensional reduced-form parameter vector. It is used to analyze the differences between Bayesian credible sets and frequentist confidence sets. We define a plug-in estimator of the identified set and show that asymptotically Bayesian highest-posterior-density sets exclude parts of the estimated identified set, whereas it is well known that frequentist confidence sets extend beyond the boundaries of the estimated identified set. We recommend reporting estimates of the identified set and information about the conditional prior along with Bayesian credible sets. A numerical illustration for a two-player entry game is provided.