Parametric Structure of Probabilities in Bayesian Networks

The paper presents a method for uncertainty propagation in Bayesian networks in symbolic, as opposed to numeric, form. The algebraic structure of probabilities is characterized. The prior probabilities of instantiations and the marginal probabilities are shown to be rational functions of the parameters, where the polynomials appearing in the numerator and the denominator are at the most first degree in each of the parameters. It is shown that numeric propagation algorithms can be adapted for symbolic computations by means of canonical components. Furthermore, the same algorithms can be used to build automatic code generators for symbolic propagation of evidence. An example of uncertainty propagation in a clique tree is used to illustrate all the steps and the corresponding code in Mathematica is given. Finally, it is shown that upper and lower bounds for the marginal probabilities of nodes are attained at one of the canonical components.