Fusion and propagation of graphical belief models: an implementation and an example
Russell G. Almond · 1990
Problems with large numbers of attributes (variables) are difficult both because of the complexity of the outcome spaces and the need to organize the component information. Graphical Models provide a useful organizational tool in the fields of artificial intelligence and risk analysis. Belief functions, based on upper and lower probabilities, provide a rich collection of models for both the interaction among attributes and information about single attributes. The fusion and propagation algorithm provides an efficient method for computing margins of joint distributions using the structural information of the graphical model. However, the lack of computational tools implementing these techniques has limited their applicability to real-world examples. The BELIEF package is an implementation of the fusion and propagation algorithm for belief functions. It allows specification of component models using PS-sets--an abbreviated notation for structured sets--and it re-organizes the graphical model into a tree model in which the fusion and propagation algorithm is implemented by message passing. In this environment, it is simple to perform sensitivity analyses on the graphical models and to diagnostically trace information (or lack of information) back to its source. A typical fault tree from a Probabilistic Risk Assessment (Spencer, Diegert and Easterling (1985), NUREG CR-2787) provides an example of graphical belief models. Graphical belief functions easily model the fault tree and the failure of both data-available and data-free components. Including dependencies among components of the same type to model common information about failure rates increases the complexity of the problem until exact calculation of the belief of system failure is intractable. In this example, because the system is coherent (by a theorem proved here) beliefs of system failure can be calculated using Monte Carlo integration to break the type dependence. The resulting beliefs and plausibilities of system failure are slightly more conservative than subjective uncertainty estimates obtained in Spencer, Diegert and Easterling (1985). Although the belief function model does not provide a point estimate of the failure rate, it does provide a more honest assessment of the lack of information about the failure of the system.