Learning Bayesian networks by constrained Bayesian estimation

Abstract

Bayesian networks (BNs) have become increasingly popular in recent years due to their wide-ranging applications in modeling uncertain knowledge. An essential problem about discrete BNs is learning conditional probability table (CPT) parameters. If training data are sparse, purely data-driven methods often fail to learn accurate parameters. Then, expert judgments can be introduced to overcome this challenge. Parameter constraints deduced from expert judgments can cause parameter estimates to be consistent with domain knowledge. In addition, Dirichlet priors contain information that helps improve learning accuracy. This paper proposes a constrained Bayesian estimation approach to learn CPTs by incorporating constraints and Dirichlet priors. First, a posterior distribution of BN parameters is developed over a restricted parameter space based on training data and Dirichlet priors. Then, the expectation of the posterior distribution is taken as a parameter estimation. As it is difficult to directly compute the expectation for a continuous distribution with an irregular feasible domain, we apply the Monte Carlo method to approximate it. In the experiments on learning standard BNs, the proposed method outperforms competing methods. It suggests that the proposed method can facilitate solving real-world problems. Additionally, a case study of Wine data demonstrates that the proposed method achieves the highest classification accuracy.

Fault diagnosis for rolling bearing based on parameter transfer Bayesian network

“…The Bayesian network is an extension of the Bayesian theory proposed by Pearl of the University of California in the late 1980s, which is used to solve the lack of an effective reasoning algorithm for multivariate joint probability density in probabilistic reasoning methods 36–39 …”

“…The Bayesian network is an extension of the Bayesian theory proposed by Pearl of the University of California in the late 1980s, which is used to solve the lack of an effective reasoning algorithm for multivariate joint probability density in probabilistic reasoning methods. [36][37][38][39] Bayes network is expressed as the form of probability 𝑄(𝐺, 𝑃). G is a directed acyclic graph with m nodes (random variables), and the edges between nodes represent the dependence relationship between the random variables.…”

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“…In the end, the BN parameters are obtained using the MAP formula. Gao [21] proposes a constrained Bayesian estimation (CBE) algorithm that enhances learning accuracy by introducing expert criteria. Di [22] proposes a constrained adjusted MAP (CaMAP) algorithm by choosing a reasonable equivalent sample size.…”

“…In addition, study [19] only applies to monotonic constraints, and study [20] only applies to approximate equality constraints. The prior distribution of [21] is set to be the BDeu priors rather than the transferred priors, which are more meaningful. When no parameter constraints are available, the CaMAP [22] is inferior to the MAP.…”

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“…Furthermore, for the incomplete data, an improved expectation maximum algorithm was proposed by combining with the CMAP method. The paper [35] proposed a constrained Bayesian estimation approach to learn parameters by incorporating constraints deduced from expert judgments and Dirichlet priors.…”