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 the most promising method seems to be the intervals of measures method. 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 expected loss, but a set of such values corresponding to all possible prior distributions. In this report 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. A simple algorithm to determine these bounds is described. The choice of a decision can be based on a comparison of the bounds of the expected loss per decision. Key words and phrases: Bayesian decision theory, imprecise probabilities, intervals of measures, lower and upper bounds for expected loss.