融合专家先验知识和单调性约束的贝叶斯网络参数学习方法

doi:10.3969/j.issn.1001-506X.2020.03.019

Abstract

Abstract:

Aiming at the Bayesian network parameter learning problem under the condition of the small sample set, a Bayesian network parameter learning method combining expert prior knowledge and monotonic constraints is proposed. This method integrates the expert priori knowledge into the process of parameter learning of Bayesian network with monotonic constraints in the form of normal distribution and further improves the accuracy and stability of parameter learning of Bayesian network under the condition of the small sample set. Simulation experiments are conducted under the condition of small sample sets, and the results show that compared with the other three main methods, the average Kullback-Leibler (KL) divergence is significantly reduced, and the running time is higher than the other three methods. Considering the learning accuracy and running time comprehensively, this method is superior to the other three methods mentioned above. The method is applied to the gas turbine health condition evaluation, and the result is consistent with the actual condition, which verifies the effectiveness of the method.

Key words: Bayesian network, parametric learning, small sample set, monotonic constraint, normal distribution

CLC Number:

TP18

Qiang ZENG, Zheng HUANG, Shuhuan WEI. Bayesian network parameter learning method based on expert priori knowledge and monotonic constraints[J]. Systems Engineering and Electronics, 2020, 42(3): 646-652.

Figures/Tables 15

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References 24

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节点	参数值
S	P_S\|C(0\|0)=0.2
	P_S\|C(1\|0)=0.8
	P_S\|C(0\|1)=0.6
	P_S\|C(1\|1)=0.4
R	P_R\|C(0\|0)=0.9
	P_R\|C(1\|0)=0.1
	P_R\|C(0\|1)=0.2
	P_R\|C(1\|1)=0.8
W	P_{W\|S, R}(0\|0, 0)=0.9
	P_{W\|S, R}(1\|0, 0)=0.1
	P_{W\|S, R}(0\|0, 1)=0.2
	P_{W\|S, R}(1\|0, 1)=0.8
	P_{W\|S, R}(0\|1, 0)=0.1
	P_{W\|S, R}(1\|1, 0)=0.9
	P_{W\|S, R}(0\|1, 1)=0.01
	P_{W\|S, R}(1\|1, 1)=0.99

编号	约束
1	P_S\|C(1\|0)>P_S\|C(0\|0)
2	P_S\|C(0\|1)>P_S\|C(1\|1)
3	P_R\|C(0\|0)>P_R\|C(1\|0)
4	P_R\|C(1\|1)>P_R\|C(1\|0)
5	P_{W\|S, R}(1\|0, 1)>P_{W\|S, R}(0\|0, 1)
6	P_{W\|S, R}(1\|1, 1)>P_{W\|S, R}(0\|1, 1)
7	P_{W\|S, R}(0\|0, 0)>P_{W\|S, R}(1\|0, 0)
8	P_{W\|S, R}(1\|1, 0)>P_{W\|S, R}(0\|1, 0)

编号	参数先验
1	P_S\|C(0\|0)≈0.2
2	P_S\|C(1\|1)≈0.4
3	P_R\|C(1\|0)≈0.1
4	P_R\|C(0\|1)≈0.2
5	P_{W\|S, R}(1\|0, 0)≈0.1
6	P_{W\|S, R}(0\|0, 1)≈0.2
7	P_{W\|S, R}(0\|1, 0)≈0.1
8	P_{W\|S, R}(0\|1, 1)≈0.01

W节点状态	5样本集				15样本集
W节点状态	MLE	无先验MAP	均匀分布方法	本文方法	MLE	无先验MAP	均匀分布方法	本文方法
P_{W\|S, R}(0\|0, 0)	1.000 0	0.666 7	0.850 4	0.902 6	1.000 0	0.800 0	0.851 9	0.907 3
P_{W\|S, R}(1\|0, 0)	0.000 0	0.333 3	0.149 6	0.097 4	0.000 0	0.200 0	0.148 1	0.092 7
P_{W\|S, R}(0\|0, 1)	0.000 0	0.333 3	0.241 5	0.195 3	0.200 0	0.285 7	0.242 5	0.200 0
P_{W\|S, R}(1\|0, 1)	1.000 0	0.666 7	0.758 5	0.804 7	0.800 0	0.714 3	0.757 5	0.800 0
P_{W\|S, R}(0\|1, 0)	0.000 0	0.200 0	0.173 6	0.092 7	0.000 0	0.166 7	0.167 9	0.090 5
P_{W\|S, R}(1\|1, 0)	1.000 0	0.800 0	0.826 4	0.907 3	1.000 0	0.833 3	0.832 1	0.909 5
P_{W\|S, R}(0\|1, 1)	0.000 0	0.333 3	0.020 0	0.010 0	0.000 0	0.200 0	0.019 5	0.009 8
P_{W\|S, R}(1\|1, 1)	1.000 0	0.666 7	0.980 0	0.990 0	1.000 0	0.800 0	0.980 5	0.990 2

W节点状态	25样本集				35样本集
W节点状态	MLE	无先验MAP	均匀分布方法	本文方法	MLE	无先验MAP	均匀分布方法	本文方法
P_{W\|S, R}(0\|0, 0)	0.750 0	0.666 7	0.847 7	0.897 0	0.888 9	0.818 2	0.851 4	0.899 5
P_{W\|S, R}(1\|0, 0)	0.250 0	0.333 3	0.152 3	0.103 0	0.111 1	0.181 8	0.148 6	0.100 5
P_{W\|S, R}(0\|1, 0)	0.125 0	0.200 0	0.222 6	0.193 9	0.100 0	0.166 7	0.211 0	0.190 0
P_{W\|S, R}(1\|1, 0)	0.875 0	0.800 0	0.777 4	0.806 1	0.900 0	0.833 3	0.789 0	0.810 0
P_{W\|S, R}(0\|0, 1)	0.250 0	0.300 0	0.154 2	0.105 9	0.300 0	0.300 0	0.206 5	0.105 9
P_{W\|S, R}(1\|0, 1)	0.750 0	0.700 0	0.845 8	0.894 1	0.700 0	0.700 0	0.750 0	0.894 1
P_{W\|S, R}(0\|1, 1)	0.000 0	0.166 7	0.019 3	0.009 7	0.000 0	0.125 0	0.019 0	0.009 5
P_{W\|S, R}(1\|1, 1)	1.000 0	0.833 3	0.980 7	0.990 3	1.000 0	0.875 0	0.981 0	0.990 5

样本集	MLE	无先验信息MAP	先验均匀分布方法	本文方法
5	0.021	0.014	0.417	0.424
15	0.022	0.014	0.430	0.433
25	0.023	0.017	0.440	0.437
35	0.025	0.017	0.445	0.443

Bayesian network parameter learning method based on expert priori knowledge and monotonic constraints

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节点状态	T₄	T₂	N₁	N₂
0	0.15	0.29	0.25	0.12
1	0.85	0.71	0.75	0.88

编号	约束
1	P_{T\|T₂, T₄}(0\|0, 0)>P_{T\|T₂, T₄}(1\|0, 0)
2	P_{T\|T₂, T₄}(1\|0, 1)>P_{T\|T₂, T₄}(0\|0, 1)
3	P_{T\|T₂, T₄}(1\|1, 0)>P_{T\|T₂, T₄}(0\|1, 0)
4	P_{T\|T₂, T₄}(1\|1, 1)>P_{T\|T₂, T₄}(0\|1, 1)
5	P_{N\|N₁, N₂}(0\|0, 0)>P_{N\|N₁, N₂}(1\|0, 0)
6	P_{N\|N₁, N₂}(0\|0, 1) > P_{N\|N₁, N₂}(1\|0, 1)
7	P_{N\|N₁, N₂}(0\|1, 0)>P_{N\|N₁, N₂}(1\|1, 0)
8	P_{N\|N₁, N₂}(1\|1, 1)>P_{N\|N₁, N₂}(0\|1, 1)
9	P_{H\|T, N}(0\|0, 0)>P_{H\|T, N}(1\|0, 0)
10	P_{H\|T, N}(1\|0, 1)>P_{H\|T, N}(0\|0, 1)
11	P_{H\|T, N}(1\|1, 0)>P_{H\|T, N}(0\|1, 0)
12	P_{H\|T, N}(1\|1, 1)>P_{H\|T, N}(0\|1, 1)

编号	约束
1	P_{T\|T₂, T₄}(1\|0, 0)≈0.2
2	P_{T\|T₂, T₄}(0\|0, 1)≈0.3
3	P_{T\|T₂, T₄}(0\|1, 0)≈0.4
4	P_{T\|T₂, T₄}(0\|1, 1)≈0.1
5	P_{N\|NN₂, N₂}(1\|0, 0)≈0.1
6	P_{N\|NN₂, N₂}(1\|0, 1)≈0.4
7	P_{N\|NN₂, N₂}(1\|1, 0)≈0.4
8	P_{N\|NN₂, N₂}(0\|1, 1)≈0.2
9	P_{H\|T, N}(1\|0, 0)≈0.1
10	P_{H\|T, N}(0\|0, 1)≈0.4
11	P_{H\|T, N}(0\|1, 0)≈0.3
12	P_{H\|T, N}(0\|1, 1)≈0.1

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节点	T₄	0		1
T₂	0	1	0	1
T	0	0.809 1	0.349 1	0.388 3	0.100 0
T	1	0.190 9	0.650 9	0.611 7	0.900 0

节点	N₂	0		1
节点	N₁	0	1	0	1
N	0	0.878 8	0.574 0	0.587 5	0.190 9
N	1	0.121 2	0.426 0	0.412 5	0.809 1

节点	T	0		1
节点	N	0	1	0	1
H	0	0.859 9	0.440 6	0.260 3	0.137 1
H	1	0.140 1	0.559 4	0.739 7	0.862 9