Multiplicative Properties in Evaluation of Decision Trees

• In attempting to address real-life decision problems, where uncertainty about data prevails, some kind of representation of imprecise information is important and several have been proposed. In particular, first-order representations, such as sets of probability measures, upper and lower probabilities, and interval probabilities and utilities of various kinds, have been suggested for enabling a better representation of the input sentences for a subsequent decision analysis. However, sometimes second-order approaches are better suited for modelling incomplete knowledge and we demonstrate how such can add important information when handling aggregations of imprecise representations, as is the case in decision trees or probabilistic networks. Based on this, we suggest a measure of belief density for such intervals. We also demonstrate important properties when operating on general distributions. The results equally apply to approaches which do not explicitly deal with second-order distributions, instead using only first-order concepts such as upper and lower bounds. While the discussion focuses on probabilistic decision trees, the results apply to other formalisms involving products of probabilities, such as probabilistic networks, and to formalisms dealing with products of interval entities such as interval weight trees in multi-criteria decision making.

Nyckelord

Decision analysis; probabilistic reasoning; second-order probabilities; imprecise probabilities; decision rules

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