Qualitative approaches to quantifying probabilistic networks

Abstract

A probabilistic network consists of a graphical representation (a directed graph) of the important variables in a domain of application, and the relationships between them, together with a joint probability distribution over the variables. A probabilistic network allows for computing any probability of interest. The joint probability distribution factorises into conditional probability distributions such that for each variable represented in the graph a distribution is specified conditional on all possible combinations of the variable's parents in the graph. Even for a moderate sized probabilistic network, thousands of probabilities need to be specified. Often the only source of probabilistic information is the knowledge and experience of experts. People, even experts, are known not be very good at assessing probabilities, and often dislike expressing their estimates as numbers. To overcome this problem, we propose two qualitative approaches to quantifying probabilistic networks. The first approach is abstracting away from probabilities by using qualitative probabilistic networks. The second approach is to allow the use of verbal expressions of probability during elicitation. In qualitative probabilistic networks, the arcs of the directed graph are augmented with signs: `+',`-', `0', and `?', indicating the direction of shift in probability for the variable at one end of the arc, given a shift in values of the variable at the other end of the arc. For example, a positive influence of variable A on variable B indicates that higher values for B become more likely given higher values for A. Qualitative probabilistic networks allow for reasoning with probabilistic networks in a qualitative way, thereby enabling us to check the robustness of the network's structure before probabilities are assessed. In addition, the qualitative signs provide constraints on the probabilities to be elicited. Qualitative networks are, however, not very expressive and therefore easily result in uninformative answers (`?'s) during reasoning. We will suggest several refinements of the formalism of qualitative probabilistic networks that enhance their expressiveness and applicability. To make probability elicitation easier on experts, we allow them to state verbal probability expressions, such as "probable" and "impossible", as well as numbers. To this end, we have augmented a vertical probability elicitation scale with verbal expressions. These expressions, and their position on the scale, are the result of several studies we conducted. The scale, together with other ingredients such as text-fragments describing the probability to be assessed and grouping of the probabilities that should sum to 1, is used in a newly designed probability elicitation method. The method provides for the elicitation of initial rough assessments. Assessments for which the outcome of the network is very sensitive can be refined using additional experts and/or the more conventional elicitation methods. Our method has been used with two experts in oncology in the construction of a probabilistic network for oesophageal carcinoma and allows us to elicit a large number of probabilities in little time. The experts felt comfortable with the method and evaluations of the resulting network have shown that it performs quite well with the rough assessments.