Non-impeding noisy-AND tree causal models over multi-valued variables

To specify a Bayesian network (BN), a conditional probability table (CPT), often of an effect conditioned on its n causes, must be assessed for each node. Its complexity is generally exponential in n. Noisy-OR and a number of extensions reduce the complexity to linear, but can only represent reinforcing causal interactions. Non-impeding noisy-AND (NIN-AND) trees are the first causal models that explicitly express reinforcement, undermining, and their mixture. Their acquisition has a linear complexity, in terms of both the number of parameters and the size of the tree topology. As originally proposed, however, they allow only binary effects and causes. This work generalizes binary NIN-AND tree models to multi-valued effects and causes. It is shown that the generalized NIN-AND tree models express reinforcement, undermining, and their mixture at multiple levels, relative to each active value of the effect. The model acquisition is still efficient. For binary variables, they degenerate into binary NIN-AND tree models. Hence, this contribution enables CPTs of discrete BNs of arbitrary variables (binary or multi-valued) to be specified efficiently through the intuitive concepts of reinforcement and undermining.