Noisy Threshold Functions for Modelling Causal Independence in Bayesian Networks∗

Causal independence modelling is a well-known method both for reducing the size of probability tables and for explaining the underlying mechanisms in Bayesian networks. Many Bayesian network models incorporate causal independence assumptions; however, only the noisy OR and noisy AND, two examples of causal independence models, are used in practice. Their underlying assumption that either at least one cause, or all causes together, give rise to an effect, however, seems unnecessarily restrictive. In the present paper a new, more flexible, causal independence model is proposed, based on the Boolean threshold function. A connection is established between conditional probability distributions based on the noisy threshold model and Poisson binomial distributions, and the basic properties of this probability distribution are studied in some depth. We present and analyse recursive methods as well as approximation and bounding techniques to assess the conditional probabilities in the noisy threshold models.