The theory of intervals of measures (DeRobertis and Hartigan, 1981) enables the use of a set of possible probability density functies for a random variable. In this paper the concept of highest density regions is generalized for situations where lack of perfect knowledge about the distribution of a random variable is represented by intervals of measures. A definition is given for an imprecise highest density region, together with a theorem that implies that such a region is equal to a highest density region for one particular probability density function.
This result can as well be used within the theory of imprecise probabilities (Walley, 1991) as for sensitivity analysis with regard to the distribution of a random variable. Also in the Bayesian theory of statistics highest density regions play an important role.