The efficient estimation of missing information in causal inverted multiway trees

摘要

Causal networks require probability values to be supplied for all possible combinations of outcomes in the cause–effect relationships implied by the network. Only then is it possible to use the existing methods for updating the information in the network to reflect new knowledge gained in a specific situation. Supplying causal information which is complete and accurate is not always possible in many applications, for example Decision Support Systems. This requirement becomes even more difficult to achieve when a single event is influenced by a large number of other events.Maximum Entropy can be used to find minimally prejudiced estimates for missing information but this approach is, in general, computationally infeasible.However, the authors have already shown that for certain special cases of causal networks such estimates can, in fact, be found in linear time.This article extends the work to causal inverted multiway trees in which any event can be influenced by any number of other events but itself only influences at most one event. In order to achieve this extension a thorough analysis of the traditional Bayesian model is undertaken to identify the large number of constraints which a valid Maximum Entropy model must satisfy. A simplified Maximum Entropy model is proposed and formal proofs that this satisfies the Bayesian properties are given.Equating the joint event probability distributions given by the Bayesian and Maximum Entropy models enables the Lagrange multipliers of the latter to be determined. This leads to an iterative tree traversal algorithm which converges to the minimally prejudiced estimates for the missing information. When this information is added to that already provided, any existing method for updating the causal network can be utilised.