Bayesian Network Parameter Learning Based on AHP-DST Fusion of Expert Prior Knowledge

doi:10.16182/j.issn1004731x.joss.24-0658

Abstract

Abstract:

Aiming at the problem of low accuracy of BN parameter learning due to the uncertainty of a single expert prior knowledge under the condition of small sample data set, a BN parameter learning method based on AHP-DST fusion expert prior knowledge was designed. The synthetic prior knowledge of experts was calculated by using the thought of analytic hierarchy process combined with the rules of evidence theory synthesis. The expert comprehensive prior knowledge was added to the normal distribution and combined with the monotonicity constraint to obtain the virtual sample information. The virtual sample information was added to the Bayesian estimation to get the network parameter estimate. The simulation results show that the KL divergence of the proposed method is always better than that of the other four methods when the sample data is small, and the running time is slightly higher than that of the other two methods. In general, the comprehensive performance of the proposed algorithm outperforms the other four methods, and it is more suitable for the case of small sample data. The proposed method is applied to the identification of the attack intention of the air target on the surface ship, and the simulation results can reflect the actual situation well, which further verifies the effectiveness and feasibility of the method.

Key words: Bayesian network, parameter learning, small dataset, analytic hierarchy process, evidence theory, monotonic constraint

CLC Number:

TP181

Chen Haiyang, Lin Hongkai, Ren Zhifang, Liu Jing, Zhang Jing. Bayesian Network Parameter Learning Based on AHP-DST Fusion of Expert Prior Knowledge[J]. Journal of System Simulation, 2025, 37(11): 2778-2792.

Figures/Tables 25

Fig. 1

Table 1

Fig. 2

Table 2

Table 3

Table 4

Cognitive judgment matrix of expert 1

专家1	$P 1$	$P 2$	$Θ$
$P 1$	1	0	5w₁
$P 2$	0	1	3w₁
$Θ$	1/5w₁	1/3w₁	1

Table 4

Table 5

Cognitive judgment matrix of expert 2

专家2	$P 1$	$P 2$	$Θ$
$P 1$	1	0	6w₂
$P 2$	0	1	2w₂
$Θ$	1/6w₂	1/2w₂	1

Table 5

Table 6

Cognitive judgment matrix of expert 3

专家3	$P 1$	$P 2$	$Θ$
$P 1$	1	0	5w₃
$P 2$	0	1	3w₃
$Θ$	1/5w₃	1/3w₃	1

Table 6

Table 7

Table 8

Constraints

编号	约束
1	$P (S = 0 \| C = 0) > P (S = 1 \| C = 0)$
2	$P (S = 0 \| C = 1) > P (S = 1 \| C = 1)$
3	$P (R = 0 \| C = 0) > P (R = 1 \| C = 0)$
4	$P (R = 0 \| C = 1) > P (R = 1 \| C = 1)$
5	$P (W = 0 \| S = 0, R = 0) > P (W = 1 \| S = 0, R = 0)$
6	$P (W = 0 \| S = 0, R = 1) > P (W = 1 \| S = 0, R = 1)$
7	$P (W = 0 \| S = 1, R = 0) > P (W = 1 \| S = 1, R = 0)$
8	$P (W = 0 \| S = 1, R = 1) > P (W = 1 \| S = 1, R = 1)$

Table 8

Fig. 3

Fig. 4

Fig. 5

Fig. 6

Fig. 7

Table 9

Table 10

Cognitive judgment matrix of Expert 1

专家1	${F 1, F 2}$	$F 3$	$Θ$
${F 1, F 2}$	1	0	5w₁
$F 3$	0	1	3w₁
$Θ$	1/5w₁	1/3w₁	1

Table 10

Table 11

Cognitive judgment matrix of Expert 2

专家2	$F 1$	$F 2$	$Θ$
$F 1$	1	0	4w₂
$F 2$	0	1	2w₂
$Θ$	1/4w₂	1/2w₂	1

Table 11

Table 12

Cognitive judgment matrix of Expert 3

专家3	$F 1$	${F 2, F 3}$	$Θ$
$F 1$	1	0	4w₃
${F 2, F 3}$	0	1	6w₃
$Θ$	1/4w₃	1/6w₃	1

Table 12

Table 13

Cognitive judgment matrix of Expert 4

专家4	$F 1$	$F 3$	$Θ$
$F 1$	1	0	5w₄
$F 3$	0	1	2w₄
$Θ$	1/5w₄	1/2w₄	1

Table 13

Table 14

Comprehensive prior knowledge of experts

编号	约束	编号	约束
1	$P T H T \| T A I (1 \| 1) ≈ 0.593 3$	12	$P T P T \| T A I (3 \| 2) ≈ 0.201 6$
2	$P T H T \| T A I (2 \| 1) ≈ 0.117 9$	13	$P O D \| T A I (1 \| 1) ≈ 0.746 6$
3	$P T H T \| T A I (3 \| 1) ≈ 0.288 8$	14	$P O D \| T A I (2 \| 1) ≈ 0.151 3$
4	$P T H T \| T A I (1 \| 2) ≈ 0.189 4$	15	$P O D \| T A I (3 \| 1) ≈ 0.102 2$
5	$P T H T \| T A I (2 \| 2) ≈ 0.692 5$	16	$P O D \| T A I (1 \| 2) ≈ 0.097 8$
6	$P T H T \| T A I (3 \| 2) ≈ 0.118 1$	17	$P O D \| T A I (2 \| 2) ≈ 0.266 8$
7	$P T P T \| T A I (1 \| 1) ≈ 0.591 4$	18	$P O D \| T A I (3 \| 2) ≈ 0.635 4$
8	$P T P T \| T A I (2 \| 1) ≈ 0.302 6$	19	$P F C C \| T A I (1 \| 1) ≈ 0.903 7$
9	$P T P T \| T A I (3 \| 1) ≈ 0.106 1$	20	$P F C C \| T A I (2 \| 1) ≈ 0.096 3$
10	$P T P T \| T A I (1 \| 2) ≈ 0.417 5$	21	$P F C C \| T A I (1 \| 2) ≈ 0.209 8$
11	$P T P T \| T A I (2 \| 2) ≈ 0.380 9$	22	$P F C C \| T A I (2 \| 2) ≈ 0.790 2$

Table 14

Table 15

Parameter constraint set

编号	约束
1	$P (T H T = 1 \| T H I = 1) > P (T H T = 3 \| T A I = 1) > P (T H T = 2 \| T A I = 1)$
2	$P (T H T = 2 \| T H I = 2) > P (T H T = 1 \| T A I = 2) > P (T H T = 3 \| T A I = 2)$
3	$P (T P T = 1 \| T H I = 1) > P (T P T = 2 \| T A I = 1) > P (T P T = 3 \| T A I = 1)$
4	$P (T P T = 1 \| T H I = 2) > P (T P T = 2 \| T A I = 2) > P (T P T = 3 \| T A I = 2)$
5	$P (O D = 1 \| T A I = 1) > P (O D = 2 \| T A I = 1) > P (O D = 3 \| T A I = 1)$
6	$P (O D = 3 \| T A I = 2) > P (O D = 2 \| T A I = 2) > P (O D = 1 \| T A I = 2)$
7	$P (F C C = 1 \| T A I = 1) > P (F C C = 2 \| T A I = 1)$
8	$P (F C C = 2 \| T A I = 2) > P (F C C = 1 \| T A I = 2)$
9	$P (T F S = 1 \| T P T = 1) > P (T F S = 2 \| T A I = 1)$
10	$P (T F S = 1 \| T P T = 2) > P (T F S = 2 \| T A I = 2)$
11	$P (T F S = 2 \| T P T = 3) > P (T F S = 1 \| T A I = 3)$

Table 15

Table 16

BN parameter learning results

节点	状态	参数学习结果
$T A I$	$T A I = 1$	$T H T 123 = (0.561 0, 0.061 0, 0.378 0)$
		$T P T 123 = (0.581 3, 0.313 0, 0.105 7)$
		$O D 123 = (0.723 6, 0.162 6, 0.113 8)$
		$F C C 12 = (0.894 3, 0.105 7)$
	$T A I = 2$	$T H T 123 = (0.232 3, 0.622 0, 0.145 7)$
		$T P T 123 = (0.409 4, 0.421 3, 0.169 3)$
		$O D 123 = (0.133 9, 0.259 8, 0.606 3)$
		$F C C 12 = (0.228 3, 0.771 7)$
$T P T$	$T P T = 1$	$T F S 12 = (0.866 4, 0.133 6)$
	$T P T = 2$	$T F S 12 = (0.608 7, 0.391 3)$
	$T P T = 3$	$T F S 12 = (0.449 3, 0.550 7)$

Table 16

Fig. 8

Fig. 9

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认知程度	数值比率
极重要	6
极重要-重要	5
重要	4
重要-较重要	3
较重要	2

编号	约束
1	$P (S = 0 \| C = 0) > P (S = 1 \| C = 0)$
2	$P (S = 0 \| C = 1) > P (S = 1 \| C = 1)$
3	$P (R = 0 \| C = 0) > P (R = 1 \| C = 0)$
4	$P (R = 0 \| C = 1) > P (R = 1 \| C = 1)$
5	$P (W = 0 \| S = 0, R = 0) > P (W = 1 \| S = 0, R = 0)$
6	$P (W = 0 \| S = 0, R = 1) > P (W = 1 \| S = 0, R = 1)$
7	$P (W = 0 \| S = 1, R = 0) > P (W = 1 \| S = 1, R = 0)$
8	$P (W = 0 \| S = 1, R = 1) > P (W = 1 \| S = 1, R = 1)$

编号	TAI	THT	TPT	OD	FCC	TFS
1	2	1	1	2	2	1
2	1	1	1	1	1	1
3	2	3	3	3	2	1
4	1	1	2	2	1	2
5	2	3	1	1	1	1
6	2	1	1	1	2	1
7	1	1	2	1	1	2
8	1	2	1	1	1	1
9	2	3	2	3	2	2
10	1	1	1	2	1	1

编号	约束	编号	约束
1	$P T H T \| T A I (1 \| 1) ≈ 0.593 3$	12	$P T P T \| T A I (3 \| 2) ≈ 0.201 6$
2	$P T H T \| T A I (2 \| 1) ≈ 0.117 9$	13	$P O D \| T A I (1 \| 1) ≈ 0.746 6$
3	$P T H T \| T A I (3 \| 1) ≈ 0.288 8$	14	$P O D \| T A I (2 \| 1) ≈ 0.151 3$
4	$P T H T \| T A I (1 \| 2) ≈ 0.189 4$	15	$P O D \| T A I (3 \| 1) ≈ 0.102 2$
5	$P T H T \| T A I (2 \| 2) ≈ 0.692 5$	16	$P O D \| T A I (1 \| 2) ≈ 0.097 8$
6	$P T H T \| T A I (3 \| 2) ≈ 0.118 1$	17	$P O D \| T A I (2 \| 2) ≈ 0.266 8$
7	$P T P T \| T A I (1 \| 1) ≈ 0.591 4$	18	$P O D \| T A I (3 \| 2) ≈ 0.635 4$
8	$P T P T \| T A I (2 \| 1) ≈ 0.302 6$	19	$P F C C \| T A I (1 \| 1) ≈ 0.903 7$
9	$P T P T \| T A I (3 \| 1) ≈ 0.106 1$	20	$P F C C \| T A I (2 \| 1) ≈ 0.096 3$
10	$P T P T \| T A I (1 \| 2) ≈ 0.417 5$	21	$P F C C \| T A I (1 \| 2) ≈ 0.209 8$
11	$P T P T \| T A I (2 \| 2) ≈ 0.380 9$	22	$P F C C \| T A I (2 \| 2) ≈ 0.790 2$

编号	约束
1	$P (T H T = 1 \| T H I = 1) > P (T H T = 3 \| T A I = 1) > P (T H T = 2 \| T A I = 1)$
2	$P (T H T = 2 \| T H I = 2) > P (T H T = 1 \| T A I = 2) > P (T H T = 3 \| T A I = 2)$
3	$P (T P T = 1 \| T H I = 1) > P (T P T = 2 \| T A I = 1) > P (T P T = 3 \| T A I = 1)$
4	$P (T P T = 1 \| T H I = 2) > P (T P T = 2 \| T A I = 2) > P (T P T = 3 \| T A I = 2)$
5	$P (O D = 1 \| T A I = 1) > P (O D = 2 \| T A I = 1) > P (O D = 3 \| T A I = 1)$
6	$P (O D = 3 \| T A I = 2) > P (O D = 2 \| T A I = 2) > P (O D = 1 \| T A I = 2)$
7	$P (F C C = 1 \| T A I = 1) > P (F C C = 2 \| T A I = 1)$
8	$P (F C C = 2 \| T A I = 2) > P (F C C = 1 \| T A I = 2)$
9	$P (T F S = 1 \| T P T = 1) > P (T F S = 2 \| T A I = 1)$
10	$P (T F S = 1 \| T P T = 2) > P (T F S = 2 \| T A I = 2)$
11	$P (T F S = 2 \| T P T = 3) > P (T F S = 1 \| T A I = 3)$