Randomized protocols for hiding private information can fruitfully be regarded as noisy channels in the information-theoretic sense, and the inference of the concealed information can be regarded as a hypothesis-testing problem. We consider the Bayesian approach to the problem, and investigate the probability of error associated to the inference when the MAP (maximum aposteriori probability) decision rule is adopted. Our main result is a constructive characterization of a convex base of the probability of error, which allows us to compute its maximum value (over all possible input distributions), and to identify upper bounds for it in terms of simple functions. As a side result, we are able to improve substantially the Hellman-Raviv and the Santhi-Vardy bounds expressed in terms of conditional entropy. We then discuss an application of our methodology to the Crowds protocol, and in particular we show how to compute the bounds on the probability that an adversary breaks anonymity.