贝叶斯网马尔可夫性的边缘化  

Marginalization of the Markov Property for Bayesian Networks

作　　者：孙婷然 孙毅[1] SUN Tingran;SUN Yi(College of Mathematics and System Science,Xinjiang University,Urumqi 830000)

机构地区：[1]新疆大学数学与系统科学学院,乌鲁木齐830000

出　　处：《系统科学与数学》2022年第12期3380-3396,共17页Journal of Systems Science and Mathematical Sciences

基　　金：新疆维吾尔自治区自然科学基金(2022D01C406);国家自然科学基金(11861064,11726629,11726630);东北师范大学应用统计教育部重点实验室开放课题(130028906)。

摘　　要：贝叶斯网作为概率论与图论相结合的产物,在对不确定复杂系统进行建模以及降低概率推理的计算复杂度等方面具有不可替代的应用价值.当在大规模贝叶斯网上进行概率推理和数据分析时,往往不需要关心全部的变量,而是在少数变量集上进行统计推断或概率推理,这就需要人们考虑边缘模型的结构信息(即分布中的条件独立信息),而边缘模型结构实质上是马尔可夫性的边缘化.考虑到贝叶斯网的边际化运算并不封闭,文章重点研究了贝叶斯网边缘模型的极小独立图问题.在借鉴无向图模型的变量消元方法以及有向图中t-可去点定义的基础上,文章提出了有向无圈图的变量消元方法,并证明消元后所得到的图恰好是边缘化掉变量集后所得到的边缘模型的极小独立图.As a product of the combination of probability theory and graph theory,Bayesian networks have irreplaceable applications in modeling uncertain complex systems and reducing the computational complexity of probabilistic inference.When we perform probabilistic inference and data analysis on large-scale Bayesian networks,we often need to care about only a small set of variables,which requires one to consider the structural information of the marginal model.Considering that the marginalization operation of Bayesian networks is not closed,we focus on the problem of finding minimal Ⅰ-maps for marginal models of Bayesian networks.Based on the variable elimination method in undirected graphical models and the definition of t-removable vertices in directed graphical models,we propose the method of variable elimination theory of directed acyclic graphs and prove that the graph obtained after variable elimination is exactly a minimal I-map for a maginal model of a Bayesian network when marginalizing over variable set.

关 键 词：贝叶斯网 有向无圈图 边缘模型 极小独立图 变量消元 

分 类 号：O157.5[理学—数学] O212.8[理学—基础数学]