Learning Bayesian network parameters under incomplete data with domain knowledge

Bayesian networks have gained increasing attention in recent years. One key issue in Bayesian networks (BNs) is parameter learning. When training data is incomplete or sparse or when multiple hidden nodes exist, learning parameters in Bayesian networks (BNs) becomes extremely difficult. Under these circumstances, the learning algorithms are required to operate in a high-dimensional search space and they could easily get trapped among copious local maxima. This paper presents a learning algorithm to incorporate domain knowledge into the learning to regularize the otherwise ill-posed problem, to limit the search space, and to avoid local optima. Unlike the conventional approaches that typically exploit the quantitative domain knowledge such as prior probability distribution, our method systematically incorporates qualitative constraints on some of the parameters into the learning process. Specifically, the problem is formulated as a constrained optimization problem, where an objective function is defined as a combination of the likelihood function and penalty functions constructed from the qualitative domain knowledge. Then, a gradient-descent procedure is systematically integrated with the E-step and M-step of the EM algorithm, to estimate the parameters iteratively 2 until it converges. The experiments with both synthetic data and real data for facial action recognition show our algorithm improves the accuracy of the learned BN parameters significantly over the conventional EM algorithm.