Likelihood Correspondence of Toric Statistical Models

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.