Learning Bayesian Networks using the Constrained Maximum a Posteriori Probability Method

Purely data-driven methods often fail to learn accurate conditional probability table (CPT) parameters of discrete Bayesian networks (BNs) when training data are scarce or incomplete. A practical and efficient means of overcoming this problem is to introduce qualitative parameter constraints derived from expert judgments. To exploit such knowledge, in this paper, we provide a constrained maximum a posteriori (CMAP) method to learn CPT parameters by incorporating convex constraints. To further improve the CMAP method, we present a type of constrained Bayesian Dirichlet priors that is compatible with the given constraints. Combined with the CMAP method, we propose an improved expectation maximum algorithm to process incomplete data. Experiments are conducted on learning standard BNs from complete and incomplete data. The results show that the proposed method outperforms existing methods, especially when data are extremely limited or incomplete. This finding suggests the potential effective application of CMAP to real-world problems. Moreover, a real facial action unit (AU) recognition case with incomplete data is conducted by applying different parameter learning methods. The results show that the recognition accuracy of respective recognition methods can be improved by the AU BN, which is trained by the proposed method.