Combining Second-Order Belief Distributions with Qualitative Statements in Decision Analysis

There is often a need to allow for imprecise statements in real-world decision analysis. Joint modeling of intervals and qualitative statements as constraint sets is one important approach to solving this problem, with the advantage that both probabilities and utilities can be handled. However, a major limitation with interval-based approaches is that aggregated quantities such as expected utilities also become intervals, which often hinders efficient discrimination. The discriminative power can be increased by utilizing second-order information in the form of belief distributions, and this paper demonstrates how qualitative relations between variables can be incorporated into such a framework. The general case with arbitrary distributions is described first, and then a computationally efficient simulation algorithm is presented for a relevant sub-class of analyses. By allowing qualitative relations, our approach preserves the ability of interval-based methods to be deliberately imprecise. At the same time, the use of belief distributions allows more efficient discrimination, and it provides a semantically clear interpretation of the resulting beliefs within a probabilistic framework.