提出新方法高效识别似然方程解在无穷远处的异常数据
Detecting Nonproperness of Likelihood Equations
- 基于代数几何构造新算法，判定似然方程在无穷远是否有解
- 实验显示效率远超已有方法，可处理更复杂模型
- 适合从事统计建模与代数统计研究者使用
给定一个代数统计模型，关键挑战是根据数据分类其对应的似然函数正临界点数量。正临界点即似然方程组的正实解。因此，确定正临界点数量等价于对似然方程组进行实根分类。似然方程组的判别簇几何上刻画了使实解数出现异常的数据。作为判别簇的重要组成部分，非正规集（nonproperness set）包含那些导致似然方程组在无穷远处存在解的数据。当数据穿过该集合时，实解数量发生变化。因此，识别非正规集在实根分类中起关键作用。本文提出一种计算似然方程组非正规集的新方法，并证明其正确性。实验表明，该方法在效率上显著优于文献中已有方法。
原文摘要 · Abstract (English)
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.
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