Second-Order Learning and Inference using Incomplete Data for Uncertain Bayesian Networks: A Two Node Example

Efficient second-order probabilistic inference in uncertain Bayesian networks was recently introduced. However, such second -order inference methods presume training over complete training data. While the expectation-maximization framework is well-established for learning Bayesian network parameters for incomplete training data, the framework does not determine the covariance of the parameters. This paper introduces two methods to compute the covariances for the parameters of Bayesian networks or Markov random fields due to incomplete data for two-node networks. The first method computes the covariances directly from the posterior distribution of parameters, and the second method more efficiently estimates the covariances from the Fisher information matrix. Finally, the implications and effectiveness of these covariances is theoretically and empirically evaluated.