发表机构
School of Mathematical Sciences, Beihang University; School of Mathematical Sciences, Peking University; Zhongguancun Laboratory（北京航空航天大学数学科学学院; 北京大学数学科学学院; 中关村实验室）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究针对代数统计模型的似然方程，开发了一种计算非正则性集的新方法，证明其正确性，且实验表明该方法比文献已知方法效率高得多，助力实根分类。
AI 中文摘要
给定代数统计模型，一个具有挑战性的问题是根据似然函数的正临界点数量对数据进行分类。正临界点是代数系统（即似然方程）的正解，因此确定正临界点数量是似然方程的实根分类问题。似然方程系统的判别簇从几何上描述了实解数量变得异常的数据。作为判别簇的关键组成部分，非正则性集收集了使得似然方程系统存在无穷远处解的数据，因此当数据穿过非正则性集时，实解数量会发生变化，确定非正则性集在实根分类中起着至关重要的作用。本研究开发了一种计算似然方程系统非正则性集的新方法，证明了该方法的正确性，并通过实验表明其比文献中已知方法的效率高得多。
英文摘要
Given an algebraic statistical model, a challenging problem is classifying the data according to the number of positive critical points of the likelihood function. The positive critical points are the positive solutions to an algebraic system, say likelihood equations. So, identifying the number of positive critical points is a real root classification problem for the likelihood equations. A discriminant variety of a likelihood-equation system geometrically describes the data for which the number of real solutions becomes unusual. As an essential component of the discriminant variety, the nonproperness set collects the data such that the likelihood-equation system has a solution at infinity. So, the number of real solutions varies when the data passes the nonproperness set, and identifying the nonproperness set plays a crucial role in the real root classification. In this work, we develop a novel method for computing nonproperness sets of likelihood-equation systems. We prove the correctness of this method. We show experimentally that it is far more efficient than the known methods in the literature.
