Bayesian inference for partially identified convex models: Is it valid for frequentist inference?

Inference on partially identified models plays an important role in econometrics. This paper proposes novel Bayesian procedures for these models when the identified set is closed and convex and so is completely characterized by its support function. We shed new light on the connection between Bayesian and frequentist inference for partially identified convex models. We construct Bayesian credible sets for the identified set and uniform credible bands for the support function, as well as a Bayesian procedure for marginal inference, where we may be interested in just one component of the partially identified parameter. Importantly, our procedure is shown to be an asymptotically valid frequentist procedure as well. It is computationally efficient, and we describe several algorithms to implement it. We also construct confidence sets for the partially identified parameter by using the posterior distribution of the support function and show that they have correct frequentist coverage asymptotically. In addition, we establish a local linear approximation of the support function which facilitates set inference and numerical implementation of our method, and allows us to establish the Bernstein-von Mises theorem of the posterior distribution of the support function.