Propagating imprecise probabilities in Bayesian networks

Often experts are incapable of providing ''exact'' probabilities; likewise, samples on which the probabilities in networks are based must often be small and preliminary, In such cases the probabilities in the networks are imprecise, The imprecision can be handled by second order probability distributions. It is convenient to use beta or Dirichlet distributions to express the uncertainty about probabilities. The problem oi how to propagate point probabilities in a Bayesian network now is transformed into the problem of how to propagate Dirichlet distributions in Bayesian networks. It is shown that the propagation of Dirichlet distributions in Bayesian networks with incomplete data results in a system of probability mixtures of beta-binomial and Dirichlet distributions. Approximate first order probabilities and their second order probability density functions are obtained by stochastic simulation. A number of properties of the propagation of imprecise probabilities are discussed by the use of examples. An important property is that the imprecision of inferences increases rapidly as new premises are added to an argument. The imprecision can be used as a pruning criterion in a network to keep the number of variables involved in an inferential argument small, Thus, imprecision may be used as an Ockam's razor in Bayesian networks.