Modeling the interactions between discrete and continuous causal factors in Bayesian networks

The theory of causal independence is frequently used to facilitate the assessment of the probabilistic parameters of discrete probability distributions of complex Bayesian networks. Although it is possible to include continuous parameters in Bayesian networks as well, such parameters could not, so far, be modeled by means of causal-independence theory, as a theory of continuous causal independence was not available. In this paper, such a theory is developed and generalized such that it allows merging continuous with discrete parameters based on the characteristics of the problem at hand. This new theory is based on the discovered relationship between the theory of causal independence and convolution in probability theory, discussed in detail for the first time in this paper. Furthermore, the new theory is used as a basis to develop a relational theory of probabilistic interactions. It is also illustrated how this new theory can be used in connection with special probability distributions.