We propose a classification of all one-dimensional discrete statistical models with maximum likelihood degree one based on their rational parametrization. We show how all such models can be constructed from members of a smaller class of 'fundamental models' using a finite number of simple operations. We introduce 'chipsplitting games', a class of combinatorial games on a grid which we use to represent fundamental models. This combinatorial perspective enables us to show that there are only finitely many fundamental models in the probability simplex $\Delta_n$ for $n\leq 4$.

翻译：本文基于有理参数化方法，对所有最大似然度为一的一维离散统计模型进行了分类。我们证明了所有此类模型均可通过有限次简单操作，从一个更小的"基本模型"类中构造得出。我们引入"筹码分裂博弈"——一类定义在网格上的组合博弈，用以表征基本模型。这一组合视角使我们得以证明：在概率单纯形 $\Delta_n$ 中，当 $n\leq 4$ 时仅存在有限个基本模型。