A method for defining conditional probability tables with link strength parameters for a Bayesian network

In a Bayesian network the state distribution of a node is conditionally independent of the set of all its nondescendents given the set of all its parents. The conditional probability table (CPT), which relates states of the parent nodes to those of a child node, includes entries for all possible combinations of the child and parent node states. Already with moderate dimensions the size of the CPT becomes so large that eliciting the required conditional probabilities from a panel of experts becomes impractical. Motivation of this paper is to make expert elicitation more feasible for populating CPTs in discrete state Bayesian networks. To achieve this, a method is presented that specifies how the CPT can be described with the aid of link strength parameters that are assigned to each link from a parent node to a child node, and that attain values from -1 to 1. In addition to aiding parameterisation of the CPT using expert knowledge, the link strength approach also provides a means to derive complete CPTs from training data when not all state combinations of parent nodes are present in the training data set. The method presented in this paper relies on the concept of the link strength introduced in Varis and Kuikka (1994) and the generalised Noisy-Or model of Srinivas (1993). A link strength with a value of one characterises a perfect one-to-one relationship between a parent node and the child node, and a link strength with a value of zero indicates that the two nodes are completely independent. A negative link strength indicates that there is a negative relationship between the two nodes. The generalised Noisy-Or model provides a methodology for constructing the CPT for the child node given its parent nodes using a set of parameters whose number is equal to the sum of parent node states. This paper combines the link strength concept and the generalised Noisy-Or model to outline a method where the number of parameters required for defining the CPT is reduced to the number of parent nodes. The link strength approach suggested in Varis and Kuikka (1994) has been modified to better suit for variables that are defined in the ratio scale statespace. In environmental studies concentrations and loads are typical examples of such variables. Then it is reasonable to assume that those states that are close to the most likely state have a higher probability of occurrence when compared to states that are far from it. To ensure that the relationship between link strength parameters and the resulting CPT is intuitively reasonable, the following properties are targeted. First, inclusion of a parent node with a link strength value of zero should give an identical CPT to the case where that node is absent. Second, equal link strength values should result in an equally strong effect on the child node. Third, the effect of a negative link strength should have the same magnitude as the effect of an equally strong positive link, but be in the opposite direction. And finally, when all link strengths are zero, the CPT should be non-informative, i.e. all probabilities are equal to the inverse of the number of states in the child node. Simple examples are presented to demonstrate how link strength values are converted into CPTs, and the resulting CPTs are discussed in the light of the properties listed above. The main contribution of this paper is to suggest a new method for describing the CPT with link strength parameters, whose number is equal to the number of parent nodes. This method is envisaged to aid in 1) defining CPTs based on the information acquired from an expert panel, and 2) deriving complete CPTs from training data when not all state combinations of parent nodes are present in the training data set.