We consider Bayesian inference about the mean of a binary variable that is subject to misclassification error. If the error probabilities are not known, or cannot be estimated, the parameter is only partially identified. For several reasonable and intuitive prior distributions of the misclassification probabilities, we derive new analytical expressions for the posterior distribution. Our results circumvent the need for Markov chain Monte Carlo simulation. The priors we use lead to regions in the identified set that are aposteriori more likely than others.