Maximum likelihood estimation (MLE) is a fundamental problem in statistics. Characteristics of the MLE problem for discrete 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 this ideal 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. All of our constructions are implemented in $\textit{Macaulay2}$ in a package $\texttt{LikelihoodGeometry}$ along with other tools of use in algebraic statistics. We end with an experimental section using these implementations on several interesting examples.

翻译：最大似然估计（MLE）是统计学中的一个基本问题。离散代数统计模型的MLE问题的特征反映在**似然对应**的几何结构中，该簇将数据与其最大似然估计量联系在一起。我们为一大类环面模型构建了这一理想，并针对由多路列联表产生的完全独立性与联合独立性模型，找到了一个Gröbner基。我们所有的构造均在 $\textit{Macaulay2}$ 软件中通过 $\texttt{LikelihoodGeometry}$ 包实现，该包还包含代数统计中其他有用的工具。最后，我们使用这些实现工具在几个有趣的例子上进行了实验分析。