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The Bayesian Network (BN) structure learning algorithm based on dynamic programming can obtain global optimal solutions. However, when the sample cannot fully contain the information of the real structure, especially when the sample size is small, the obtained structure is inaccurate. Therefore, this paper studies the planning mode and connotation of dynamic programming, restricts its process with edge and path constraints, and proposes a dynamic programming BN structure learning algorithm with double constraints under small sample conditions. The algorithm uses double constraints to limit the planning process of dynamic programming and reduces the planning space. Then, it uses double constraints to limit the selection of the optimal parent node to ensure that the optimal structure conforms to prior knowledge. Finally, the integrating prior-knowledge method and the non-integrating prior-knowledge method are simulated and compared. The simulation results verify the effectiveness of the method proposed and prove that the integrating prior knowledge can significantly improve the efficiency and accuracy of BN structure learning.

Maximum a posteriori estimation (MAP) with Dirichlet prior has been shown to be effective in improving the parameter learning of Bayesian networks when the available data are insufficient. Given no extra domain knowledge, uniform prior is often considered for regularization. However, when the underlying parameter distribution is non-uniform or skewed, uniform prior does not work well, and a more informative prior is required. In reality, unless the domain experts are extremely unfamiliar with the network, they would be able to provide some reliable knowledge on the studied network. With that knowledge, we can automatically refine informative priors and select reasonable equivalent sample size (ESS). In this paper, considering the parameter constraints that are transformed from the domain knowledge, we propose a Constrained adjusted Maximum a Posteriori (CaMAP) estimation method, which is featured by two novel techniques. First, to draw an informative prior distribution (or prior shape), we present a novel sampling method that can construct the prior distribution from the constraints. Then, to find the optimal ESS (or prior strength), we derive constraints on the ESS from the parameter constraints and select the optimal ESS by cross-validation. Numerical experiments show that the proposed method is superior to other learning algorithms.