Propagation techniques in probabilistic expert systems
Techniques for the construction of probabilistic expert systems comprising both discrete and continuous random variables are presented. In particular we are concerned with how continuous random variables may be incorporated into an expert system - an area which has previously received relatively little attention. We investigate and extend the numeric techniques of other authors, and develop two new approaches. The first approach makes use of computer algebra. This exact technique enables a probability distribution to be expressed and manipulated in terms of its algebraic formula resulting in no loss of information. Our second approach is an approximate method based upon cubic spline interpolation. We constrain the probability density function of a continuous variable to a finite set of points at which we have both function values and first derivatives. These values may then be held in a potential table and treated in an almost identical fashion to discrete variables. While symbolic techniques are shown to be only appropriate in special cases, cubic spline interpolation, though less accurate, is widely applicable. We combine these techniques to form a hybrid methodology in which discrete variables, symbolic continuous variables, and spline interpolated continuous variables may exist not only in the same junction tree, but also in the same universe. We show how propagation algorithms may be constructed for these various cases and investigate how the means, variances and probability density functions of the marginal distributions in the system may be generated. It is shown how evidence of either a numeric or a symbolic nature may be incorporated into such systems and how simulation studies may be performed. The techniques we develop are implemented in the computer language Mathematica and an outline of how this may be accomplished is presented.