Bounds for expected loss in Bayesian decision theory with imprecise prior probabilities

Classical Bayesian inference uses the expected value of a loss function with regard to a single prior distribution for a parameter to compare decisions, and an optimal decision minimizes the expected loss. Recently interest has grown in generalizations of this framework without specified priors, to allow imprecise prior probabilities. Within the Bayesian context a promising method is the concept of intervals of measures. A major problem for the application of this method to decision problems seems to be the amount of calculation required, since for each decision there is no single value for the expected loss, but a set of such values corresponding to all possible prior distributions. In this paper the determination of lower and upper bounds for such a set of expected loss values with regard to a single decision is discussed, and general results are derived which show that the situation is less severe than would be expected at first sight. The choice of a decision can be based on a comparison of the bounds of the expected loss per decision.