This paper advances the use of the ranked nodes method (RNM) to portray probabilistic relationships of continuous quantities in Bayesian networks (BNs). In RNM, continuous quantities are represented by ranked nodes with discrete ordinal scales. The probabilistic relationships of the nodes are quantified in conditional probability tables (CPTs) generated with expert-elicited parameters. When ranked nodes are formed by discretizing continuous scales, ignorance about the functioning of RNM can lead to discretizations that make the generation of sensible CPTs impossible. While a guideline exists on this matter, it is limited by a requirement to define an equal number of ordinal states for all the nodes. This paper presents two novel discretization approaches that consider the functioning of RNM and allow the nodes to have non-equal numbers of ordinal states. In the first one, called the “static discretization approach”, the nodes can be given any desired discretizations that stay unchanged during the use of the BN. In the second one, called the “dynamic discretization approach”, the discretizations are algorithmically updated during the use of the BN to help manage the sizes of the generated CPTs. Both approaches are based on the original idea that, besides the RNM parameters, the nodes probabilistic relationship is defined by initial RNM-compatible discretizations elicited from the domain expert. Overall, the new approaches offer an easier and more versatile way of using RNM to depict the probabilistic relationships of continuous quantities. In doing so, they also facilitate the effective and diverse use of BNs in decision support systems.

This paper provides new insight into the theoretical principle and the practical applicability of the ranked nodes method (RNM) that is used to construct conditional probability tables (CPTs) for Bayesian networks (BNs) by expert elicitation. RNM is designed for specific types of discrete random variables called ranked nodes that are common in real-world applications of BNs. Despite its active use in recent years, there remains ambiguity about the exact theoretical basis of RNM which can hamper its effective employment. In addition, there are a lack of studies about the general ability of CPTs generated with RNM to represent probabilistic relationships in real-world applications. In this paper, it is shown how the generation of probabilities with RNM is underpinned by a regression model of continuous random variables. Then, it is experimentally determined that in typical applications of RNM, one can generate in a matter of seconds CPTs whose elements reflect well probabilities given by the underlying regression model. Another experiment discovers that CPTs generated with RNM provide a good average fit to a large portion of various real-world CPTs investigated. This confirms the usefulness of RNM in practical applications. The results of the experiment also indicate that choices made by the user of RNM can considerably impact the ability of a generated CPT to represent a given probabilistic relationship. This paper then provides practical advice on the efficient use of RNM with regard to the user-controlled features explored in the experiment.

This paper elaborates on the ranked nodes method (RNM) that is used for constructing conditional probability tables (CPTs) for Bayesian networks consisting of a class of nodes called ranked nodes. Such nodes typically represent continuous quantities that lack well-established interval scales and are hence expressed by ordinal scales. Based on expert elicitation, the CPT of a child node is generated in RNM by aggregating weighted states of parent nodes with a weight expression. RNM is also applied to nodes that are expressed by interval scales. However, the use of the method in this way may be ineffective due to challenges which are not addressed in the existing literature but are demonstrated through an illustrative example in this paper. To overcome the challenges, the paper introduces a novel approach that facilitates the use of RNM. It consists of guidelines concerning the discretization of the interval scales into ordinal ones and the determination of a weight expression and weights based on assessments of the expert about the mode of the child node. The determination is premised on interpretations and feasibility conditions of the weights derived in the paper. The utilization of the approach is demonstrated with the illustrative example throughout the paper.