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 \textit{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\"{o}bner 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：极大似然估计是统计学中的基本问题。代数统计模型中极大似然估计问题的特性反映在\textit{似然对应关系}的几何结构上——这是一种将数据与其极大似然估计相联系的对象。我们为一大类环面模型构造了似然对应关系的理想，并在多维列联表产生的完全独立与联合独立模型情形下找到了其Gröbner基。这些结果不仅揭示了此类模型的内在性质，还为求解极大似然估计问题提供了更高效的计算策略。