Efficient decomposition of Bayesian networks with non-graded variables

Elicitation, estimation and exact inference in Bayesian Networks (BNs) are often difficult because the dimension of each Conditional Probability Table (CPT) grows exponentially with the increase in the number of parent variables. The Noisy-MAX decomposition has been proposed to break down a large CPT into several smaller CPTs exploiting the assumption of causal independence, i.e., absence of causal interaction among parent variables. In this way, the number of conditional probabilities to be elicited or estimated and the computational burden of the joint tree algorithm for exact inference are reduced. Unfortunately, the Noisy-MAX decomposition is suited to graded variables only, i.e., ordinal variables with the lowest state as reference, but real-world applications of BNs may also involve a number of non-graded variables, like the ones with reference state in the middle of the sample space (double-graded variables) and with two or more unordered non-reference states (multi-valued nominal variables). In this paper, we propose the causal independence decomposition, which includes the Noisy-MAX and two generalizations suited to double-graded and multi-valued nominal variables. While the general definition of BN implicitly assumes the presence of all the possible causal interactions, our proposal is based on causal independence, and causal interaction is a feature that can be added upon need. The impact of our proposal is investigated on a published BN for the diagnosis of acute cardiopulmonary diseases.