Bridging uncertain and ambiguous knowledge with imprecise probabilities

Model-based environmental decision support requires that uncertainty be rigorously evaluated. Whether uncertainty is aleatory or epistemic, we argue that probability is the natural mathematical construct for describing uncertainty in predictions used for decision-making. If expert knowledge is elicited using stated preferences between lotteries, and the experts are rational in the sense of avoiding sure loss, then the resulting knowledge quantifications will be consistent with the axiomatic foundation of probability theory. This idea can be extended to the description of intersubjective knowledge when the intent is to characterize the state of knowledge of the scientific community. Many methods for probability elicitation have been reported, but there is nearly always some degree of ambiguity in translating elicited quantities into probabilistic description. This would include: any lack of fit of a particular distributional form to elicited data; incertitude in the elicited data themselves; and/or disagreement in the elicited data across multiple experts. By replacing a precise probability distribution by a set of distributions, the mathematical concept of imprecise probabilities provides a means for representing this ambiguity. In this way, imprecise probabilities can form a bridge between total ignorance and precisely characterized risk by allowing for a continuous degree of imprecision to represent ambiguity. We introduce three metrics to describe the relative ambiguity of important attributes of probability distributions, namely their width, shape, and mode. These metrics are applicable to sets of distributions characterized by using any available method, and we derive the specific forms of these metrics for the Density Ratio Class, which we have found to have many desirable properties. Based on these metrics and on elicitation data from the literature, we use three examples to demonstrate the wide variety of ambiguity that can be present in elicited knowledge. Imprecise probabilities allow us to quantify this ambiguity and consider it in environmental decision-making. Our examples were implemented using a package we recently developed and made freely available for the R statistical programming environment. (C) 2011 Elsevier Ltd. All rights reserved.