Application of the Bayes' formula leaves little room for representation of ignorance and vagueness in quantitative estimates. Adhering to the classical probability calculus, the Bayesian approach can only replace ignorance with indifference. Shafer's belief functions are different in this respect. Free from the additivity requirement of classical probabilities, they preserve the vagueness of subjective beliefs. Together with Dempster's combination rule the belief functions offer an alternative to the Bayesian updating of probability estimates. In this paper the two methods are compared in a risk analysis application. While the results given by the Dempster-Shafer theory are, in essence, similar to those from the Bayesian analysis, the new method offers some presentational advantages for both the input and output data.