*David Sundgren, Mats Danielson, Love Ekenberg*

In real-life decision analysis, the probabilities and values of consequences are in general vague and imprecise. One way to model imprecise probabilities is to represent a probability with the interval between the lowest possible and the highest possible probability, respectively. However, there are disadvantages with this approach, one being that when an event has several possible outcomes, the distributions of belief in the different probabilities are heavily concentrated to their centers of mass, meaning that much of the information of the original intervals are lost. Representing an imprecise probability with the distribution's center of mass therefore in practice gives much the same result as using an interval, but a single number instead of an interval is computationally easier and avoids problems such as overlapping intervals. Using this, we demonstrate why second-order calculations can add information when handling imprecise representations, as is the case of decision trees or probabilistic networks. We suggest a measure of belief density for such intervals. We also demonstrate important properties when operating on general distributions. The results herein apply also to approaches which do not explicitly deal with second-order distributions, instead using only first-order concepts such as upper and lower bounds.

*Subjects:* 15.5 Decision Theory

*Submitted:* Feb 13, 2006

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