Abstract

Introducing parameter constraints has become a mainstream approach for learning Bayesian network parameters with small datasets. The QMAP (Qualitative Maximum a Posteriori) estimation has produced the best learning accuracy among existing learning approaches. However, the rejection-acceptance sampling employed in the QMAP algorithm for determining average BN(Bayesian Network) parameter values is time-consuming, particularly when the number of parameter constraints is large. This paper proposes a new analytical approach that enhances the learning efficiency of the QMAP algorithm without reducing its learning accuracy by treating the average value of the parameters as the center point of the constrained parameter region, which is a much more efficient method than the rejection-acceptance sampling method employed in the traditional QMAP algorithm. First, a novel objective function is designed and a constrained objective optimization model is constructed based on parameter constraints. Second, the constructed model is employed to obtain the center point of the constrained parameter region based on its boundary points, and the average parameter value is the average of all boundary points. The results obtained from a large number of simulation experiments with four benchmark Bayesian networks demonstrate that of the parameter learning accuracy of the proposed algorithm is slightly better than that the original QMAP algorithm under specific conditions, and the computational efficiency is substantially increased under all conditions.

Highlights

  • The above results indicate that the proposed constrained maximum a posteriori (CMAP)-I and CMAP-II algorithms have higher computational efficiencies than the qualitative maximum a posteriori (QMAP) algorithm under all conditions considered, and the learning accuracy of the CMAP-II algorithm is only slightly less than that of the QMAP algorithm when cross-distribution constraints are included within the constraint set

  • The rejection-acceptance sampling method employed in the QMAP algorithm for determining average Bayesian network (BN) parameter values is time-consuming, when the number of parameter constraints is large

  • Experiments involving four benchmark BNs demonstrated that the proposed CMAP-I and CMAP-II algorithms have higher computational efficiency than the QMAP algorithm under all experimental conditions considered, and the learning accuracy of the CMAP-II algorithm is only slightly less than that of the QMAP algorithm when cross-distribution constraints are included within the constraint set

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Summary

INTRODUCTION

Bayesian network (BN) is a statistical model that represents a group of variables as nodes in a directed acyclic graph, and denotes the conditional dependencies between variables as the graph edges. This condition can severely limit the feasibility of BNs. numerous BN modeling methods based on small datasets have been developed, and the introduction of parameter constraints has become a mainstream approach applied to address the condition. Among the single constraint type methods, Witting [15] proposed a constrained BN parameter learning algorithm that employed cross-distribution parameter constraints. The QMAP algorithm is an a posteriori estimation approach that incorporates both quantitative data and qualitative constraints This approach requires that a sufficient number of parameters be sampled from within the constrained parameter region using a rejection-acceptance sampling strategy to obtain the mean values of the parameters. The results obtained for a large number of simulation experiments with four benchmark networks demonstrate that the parameter learning accuracy of the proposed algorithm is slightly better than that of the original QMAP algorithm under specific conditions, and the computational efficiency is substantially increased under all conditions

BAYESIAN NETWORKS
B PARAMETER LEARNING IN A BAYESIAN NETWORK
C SAMPLE COMPLEXITY OF BN PARAMETER
D COMMON PARAMETER CONSTRAINTS
ANALYSIS OF LOW-DIMENSIONAL CONSTRAINED
EXPERIMENTS
Findings
CONCLUSIONS
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