概率图模型学习及其在数据分析中的应用研究

Probabilistic graphical models which are modeling tools of complicated uncertainty systems, can visually represent causal and complex conditional independent relationships among variables by the graph structure and the corresponding probability distribution. They have been successfully applied to wide range of tasks, such as machine learning, bioinformatics, financial analysis and the forecast. This project aims at solving the following problems about probabilistic graphical models. For the structural learning of graphical models, such as Bayesian networks and chain graphs, skeleton structures and statistical properties of Markov equivalence will be given. Based on them, the decomposition rules will be designed which will not violate the independent conditions. Furthermore, a problem of searching for the whole structure will be split into the recovery of local causal relationships. We will theoretically prove that the local statistics information of each variable can not be destroyed. In addition, to overcome the drawbacks of classification from small samples and multi-dimensional data with complex structure, this project will study how to transform the graphical models into the corresponding constraint optimization models. Thus, the intelligent optimization algorithms will be used for structure learning of classifiers. We will give theoretical analysis and computer simulations for the algorithms. In short, the project involves probability statistics, graph theory, the theory of intelligent optimization, and so on. The research results will promote the development of applied probability statistics and computer science.

概率图模型是复杂不确定系统建模的重要工具，它通过一个拓扑图结构和相应的概率分布来直观地表示多个变量间的条件独立关系和因果关系，从而将复杂的高维系统进行分解简化，已成功地应用于机器学习、生物信息学、金融分析与预测等多个领域．基于此，本项目拟研究如下内容：针对贝叶斯网络、链图等概率图模型的结构学习问题，给出几类图模型Markov等价的统计性质和图形刻画，以此设计出不破坏条件独立关系的分解准则；研究如何在不违背条件独立性的前提下，利用模型的Markov等价性和可分解性对复杂结构进行局部学习；针对小样本数据和多维数据分类问题，研究如何将图模型转化为相应的约束优化模型，利用智能优化算法进行分类器结构学习，并给出理论证明和仿真实验分析．本项目涉及概率统计、图论、智能优化理论等数学领域，问题的解决对应用概率统计和计算机科学的发展有着较大的促进作用．

概率图模型是复杂不确定系统建模的重要工具，它通过一个拓扑图结构和相应的概率分布来直观地表示多个变量间的条件独立关系和因果关系，从而将复杂的高维系统进行分解简化，已成功地应用于机器学习、生物信息学、金融分析与预测等多个领域．本项目针对贝叶斯网络、链图等概率图模型的结构学习问题，给出几类图模型Markov等价的统计性质和图形刻画，以此设计出不破坏条件独立关系的分解准则；研究了在不违背条件独立性的前提下，利用模型的Markov等价性和可分解性对复杂结构进行局部学习；针对小样本数据和多维数据分类问题，研究了在不违背条件独立性的前提下，利用智能优化算法进行分类器结构学习，并给出了理论证明和仿真实验分析．项目涉及概率统计、图论、智能优化理论等数学领域，问题的解决对应用概率统计和计算机科学的发展有着较大的促进作用．基于以上研究，撰写论文10余篇，其中7篇论文发表在国际重要期刊上，均被SCI检索．