Maximum likelihood estimation (MLE) is a fundamental problem in statistics. Characteristics of the MLE problem for algebraic statistical models are reflected in the geometry of the likelihood correspondence, a variety that ties together data and their maximum likelihood estimators. We construct the ideal of the likelihood correspondence for the large class of toric models and find a Gr\"obner basis in the case of complete and joint independence models arising from multi-way contingency tables. These results provide insight into their properties and offer faster computational strategies for solving the MLE problem.

翻译：极大似然估计（MLE）是统计学中的基本问题。代数统计模型中MLE问题的特性体现在似然对应关系的几何结构上，这种结构将数据与其极大似然估计量联系在一起。本文针对一大类环面模型构建了似然对应关系的理想，并在由多维列联表导出完全独立模型与联合独立模型的情形下找到了其Gröbner基。这些结果不仅揭示了此类模型的性质，还为求解MLE问题提供了更高效的计算策略。