We study the problem of maximum likelihood (ML) estimation for statistical models defined by reflexive polytopes. Our focus is on the maximum likelihood degree of these models as an algebraic measure of complexity of the corresponding optimization problem. We compute the ML degrees of all 4319 classes of three-dimensional reflexive polytopes, and observe some surprising behavior in terms of the presence of gaps between ML degrees and degrees of the associated toric varieties. We interpret these drops in the context of discriminants and prove formulas for the ML degree for families of reflexive polytopes, including the hypercube and its dual, the cross polytope, in arbitrary dimension. In particular, we determine a family of embeddings for the $d$-cube that implies ML degree one. Finally, we discuss generalized constructions of families of reflexive polytopes in terms of their ML degrees.

翻译：我们研究了由自反多面体定义的统计模型的最大似然估计问题。我们的重点在于这些模型的最大似然度，将其作为相应优化问题复杂性的代数度量。我们计算了所有4319类三维自反多面体的最大似然度，并观察到最大似然度与相关环簇次数之间出现间隔的一些意外现象。我们在判别式的背景下解释这些下降，并证明了自反多面体族的最大似然度公式，包括任意维度的超立方体及其对偶——交叉多面体。特别地，我们确定了一个$d$维立方体的嵌入族，该族意味着最大似然度为1。最后，我们讨论了基于最大似然度的自反多面体族的广义构造。