基于模糊约束的贝叶斯网络参数学习

doi:10.12305/j.issn.1001-506X.2023.02.15

摘要/Abstract

摘要：

针对小样本下贝叶斯网络参数学习结果不准确的问题, 提出一种模糊最大后验估计方法, 该方法将模糊理论引入到参数学习中, 通过对约束效力的度量, 利用隶属度函数来确定超参进行学习, 以提高约束使用的准确性。实验证明, 所提方法可以有效提高参数学习的精度。除此之外, 将所提方法应用到网络安全评估中, 将通用漏洞评分系统作为专家先验参数, 结合漏洞信息迁移样本来进行参数学习。最后, 通过节点和路径安全评估验证了所提方法的有效性。

关键词: 贝叶斯网络, 隶属度函数, 参数学习, 网络安全评估

Abstract:

To address the problem of Bayesian networks parameter learning under small datasets, a fuzzy maximum posteriori estimation method is proposed, introducing fuzzy theory into parameter learning. The hyperparameter is determined by using the membership function to measure constraint effectiveness to improve the accuracy of constraint usage for learning. Experiments prove that the proposed method can effectively improve the accuracy of parameter learning. In addition, the proposed parameter learning method is applied to a network security assessment by using common vulnerability scoring system as expert priori parameters and combining vulnerability transfer samples to perform parameter learning. Finally, the node and path security evaluation verifies the effectiveness of the proposed algorithm.

Key words: Bayesian network, membership function, parameter learning, network security assessment

中图分类号:

TP181

茹鑫鑫, 高晓光, 王阳阳. 基于模糊约束的贝叶斯网络参数学习[J]. 系统工程与电子技术, 2023, 45(2): 444-452.

Xinxin RU, Xiaoguang GAO, Yangyang WANG. Bayesian network parameter learning based on fuzzy constraints[J]. Systems Engineering and Electronics, 2023, 45(2): 444-452.

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编号	约束
1	0.9≤P(V₁=低)≤1
2	P(V₂=真)＜P(V₂=假)
3	0.9≤P(V₃=假\|V₁=低, V₂=真)≤1
4	0.9≤P(V₃=假\|V₁=高, V₂=真)≤1
5	0.9≤P(V₃=假\|V₁=低, V₂=假)≤1
6	0.9≤P(V₃=假\|V₁=高, V₂=假)≤1
7	0.9≤P(V₄=阳性\|V₃=真)≤1
8	P(V₄=阳性\|V₃=假)＜P(V₄=阴性\|V₃=假)
9	P(V₅=真\|V₃=真)＞P(V₅=假\|V₃=真)
10	P(V₅=真\|V₃=假)＜P(V₅=假\|V₃=假)

BNs	MLE	MAP	CMLE	QMAP	FMAP
Earthquake	1.173 3±0.766 6	0.348 1±0.057 5	0.044 2±0.022 7	0.105 4±0.047 8	0.080 5±0.032 8
Cancer	0.846 7±0.744 7	0.844 0±0.097 4	0.104 2±0.035 4	0.052 9±0.043 1	0.049 8±00.000 1
Sachs	0.983 2±0.235 8	0.392 1±0.024 4	0.352 6±0.099 5	0.198 3±0.027 8	0.147 3±0.010 3
Boerlage92	1.217 9±0.216 1	0.332 7±0.022 5	0.143 8±0.088 2	0.142 3±0.024 2	0.072 9±0.014 8
Win95pts	0.582 8±0.110 6	0.329 9±0.007 2	0.293 9±0.057 0	0.233 0±0.013 0	0.213 3±0.004 5
Andes	1.005 2±0.067 4	0.394 2±0.005 6	0.161 9±0.020 5	0.172 8±0.009 5	0.132 5±0.004 1
均值	0.968 2	0.440 2	0.183 4	0.150 8	0.116 1

BNs	MLE	MAP	CMLE	QMAP	FMAP
Earthquake	0.936 7±0.898 3	0.406 6±0.065 0	0.014 8±0.011 4	0.121 5±0.049 2	0.060 0±0.019 7
Cancer	0.470 2±0.442 8	0.416 5±0.083 4	0.059 9±0.032 1	0.036 7±0.017 9	0.020 6±00.001 0
Sachs	0.603 3±0.155 3	0.430 9±0.029 8	0.183 4±0.080 0	0.126 2±0.019 4	0.089 8±0.008 4
Boerlage92	0.510 7±0.188 7	0.365 7±0.020 0	0.030 2±0.034 5	0.076 2±0.020 5	0.023 9±0.005 7
Win95pts	0.653 8±0.086 7	0.350 5±0.004 8	0.381 1±0.060 2	0.212 7±0.012 5	0.173 6±0.004 8
Andes	0.698 2±0.051 1	0.429 6±0.005 4	0.159 5±0.016 7	0.118 7±0.007 0	0.064 7±0.002 9
均值	0.645 5	0.416 7	0.138 2	0.115 3	0.072 1

BNs	MLE	MAP	CMLE	QMAP	FMAP
Earthquake	0.377 4±0.364 4	0.431 5±0.058 2	0.014 2±0.016 2	0.083 4±0.028 4	0.056 6±0.014 1
Cancer	0.285 2±0.302 4	0.179 3±0.071 6	0.050 1±0.031 4	0.030 9±0.021 8	0.008 0±0.000 1
Sachs	0.437 3±0.106 2	0.477 8±0.019 8	0.136 1±0.038 4	0.093 7±0.013 4	0.065 4±0.007 4
Boerlage92	0.259 4±0.103 2	0.369 3±0.013 8	0.009 3±0.003 2	0.041 7±0.012 9	0.010 4±0.002 5
Win95pts	0.722 9±0.092 3	0.377 2±0.004 4	0.453 4±0.058 6	0.207 5±0.014 0	0.153 2±0.002 3
Andes	0.466 3±0.038 0	0.447 4±0.006 2	0.109 1±0.015 8	0.083 1±0.005 4	0.039 3±0.002 5
均值	0.424 8	0.380 5	0.128 7	0.090 1	0.055 5

BNs	CMLE	QMAP	FMAP
Earthquake	0.834 2± 0.563 1	0.117 0± 0.041 1	0.073 8± 0.041 7
Cancer	0.578 4± 0.355 7	0.075 3± 0.054 0	0.054 7± 0.040 5
Sachs	0.679 7± 0.138 5	0.194 3± 0.019 2	0.144 6± 0.008 4
Boerlage92	0.911 8± 0.216 6	0.153 2± 0.029 8	0.075 4± 0.011 8
Win95pts	0.352 2± 0.055 6	0.226 4± 0.009 7	0.214 7± 0.004 6
Andes	0.476 9± 0.045 5	0.190 3± 0.009 7	0.133 7± 0.003 1
均值	0.638 9	0.159 4	0.116 2