Bayesian Networks: Aspects of Approximate Inference

Abstract

A Bayesian network can be used to model consisely the probabilistic knowledge with respect to a given problem domain. Such a network consists of an acyclic directed graph in which the nodes represent stochastic variables, supplemented with probabilities indicating the strength of the influences between neighbouring variables. A qualitative probabilistic network is an abstraction of a Bayesian network in which the probabilistic influences among the variables are modelled by means of signs. A non-monotonic influence between two variables is associated with the ambiguous sign '?', which indicates that the actual sign of the influence depends on the state of the network. The presence of such ambiguous signs is undesirable as it tends to lead to uninformative results upon inference. In each specific state of the network, however, the influence between two variables is unambiguous. Now to capture the current effect of the influence this thesis introduces the concept of situational sign. It is shown how situational signs can be used upon inference and how they are updated as the state of the network changes. By means of a real-life qualitative network in oncology it is demonstrated that the use of situational signs can effectively forestall uninformative results upon inference. The loopy-propagation algorithm provides for approximate inference with a Bayesian network. Upon loopy propagation errors may arise in the computed probabilities due to the presence of loops in the graph. This thesis indicates that two different types of error arise in the computed probabilities which are termed convergence errors and cycling errors. These types of error are investigated in more detail for the nodes with two or more incoming arcs from a loop and for the other loop nodes seperately. For nodes with two or more incoming arcs on the loop a general expression is derived for the error that is found in the probabilities computed for these nodes in a network in its prior state. This expression includes a weighting factor that is captured by the newly defined notion of quantitative parental synergy. For the other loop nodes, the effect of the cycling error on the decisiveness of the computed probabilities is analysed. More specifically, the over- or underconfidence of these approximations is linked to two concepts from qualitative probabilistic networks. The thesis concludes with an analysis of an algorithm for inference with undirected networks which is equivalent to the loopy-propagation algorithm. It is shown how, although in undirected networks all errors arise from the cycling of information, the convergence error is embedded in the algorithm.