An Algorithm for Inferences in a Polytree with Heterogeneous Conditional Distributions

This paper describes a general scheme for accomodating different types of conditional distributions in a Bayesian network. The algorithm is based on the polytree algorithm for Bayesian network inference, in which “messages” (probability distributions and likelihood functions) are computed. The posterior for a given variable depends on the messages sent to it by its parents and children, if any. In this scheme, an exact result is computed if such a result is known for the incoming messages, otherwise an approximation is computed, which is a mixture of Gaussians. The approximation may then be propagated to other variables. Approximations for likelihood functions (λ-messages) are not computed; the approximation step is put off until the likelihood function is combined with a probability distribution — this avoids certain numerical difficulties. In contrast with standard polytree algorithms, which can only accomodate distributions of a few types at most, this heterogeneous polytree algorithm can, in principle, handle any kind of continuous or discrete conditional distribution. With standard algorithms, it is necessary to construct an approximate Bayesian network, in which one then computes exact results; the heterogeneous polytree algorithm, on the other hand, computes approximate results in the original Bayesian network. The most important advantage of the new algorithm is that the Bayesian network can be directly represented using the conditional distributions most appropriate for the problem domain.