A Bayesian analysis of a nonidentified model is always possible if a proper prior on all the parameters is specified. There is, however, no Bayesian free lunch. The “price” is that there exist quantities about which the data are uninformative, i.e., their marginal prior and posterior distributions are identical. In the case of improper priors the analysis is problematic—resulting posteriors can be improper. This study investigates both proper and improper cases through a series of examples.