Deduction from Conditional Knowledge on Bayesian Networks with Interval Probability Parameters

We propose a Bayesian higher-order probability logic reasoning approach with interval probability parameters to the problem of making inference from conditional knowledge, which combines weak conditional probability and conditional event algebra for approximate inferences. We define the bound-limited weak conditional interval probabilities, the corresponding probabilistic description, and the multiplication rules of weak conditional probabilities for joint probability distribution, and use higher-order conditional event to resolve a discrepancy between logic and probability. By extending normal measurable space with conditional event, we bring logic consistent with probability in denoting conditional knowledge, and then transform a higher-order conditional event to normal events and corresponding logical joint events via conditional event algebra. Based on multiplication rules, we compute the quantitative values of the events with interval parameters, and evaluate the value of higher-order conditional event and finish reasoning process. An illustrative example of application of our method shows how we make inferences from conditional knowledge.