The purpose of this work is to provide theoretical foundations of, as well as some computational aspects on, a theory for analysing decisions under risk, when the available information is vague and imprecise. Many approaches to model unprecise information, e.g., by using interval methods, have prevailed. However, such representation models are unnecessarily restrictive since they do not admit discrimination between beliefs in different values, i.e., the epistemologically possible values have equal weights. In many situations, for instance, when the underlying information results from learning techniques based on variance analyses of statistical data, the expressibility must be extended for a more perceptive treatment of the decision situation. Our contribution herein is an approach for enabling a refinement of the representation model, allowing for an elaborated discrimination of possible values by using belief distributions with weak restrictions. We show how to derive admissible classes of local distributions from sets of global distributions and introduce measures expressing into which extent explicit local distributions can be used for modelling decision situations. As will turn out, this results in a theory that has very attractive features from a computational viewpoint.
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.
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