In this paper we, like several studies in the recent literature, employ a Bayesian approach to estimation and inference in models with endogeneity concerns by imposing weaker prior assumptions than complete excludability. When allowing for instrument imperfection of this type, the model is only partially identified, and as a consequence standard estimates obtained from the Gibbs simulations can be unacceptably imprecise. We thus describe a substantially improved semi-analytic method for calculating parameter marginal posteriors of interest that only require use of the well-mixing simulations associated with the identifiable model parameters and the form of the conditional prior. Our methods are also applied in an illustrative application involving the impact of body mass index on earnings.