We investigate the geometry of a family of log-linear statistical models called quasi-independence models. The toric fiber product is useful for understanding the geometry of parameter inference in these models because the maximum likelihood degree is multiplicative under the TFP. We define the coordinate toric fiber product, or cTFP, and give necessary and sufficient conditions under which a quasi-independence model is a cTFP of lower-order models. We show that the vanishing ideal of every 2-way quasi-independence model with ML-degree 1 can be realized as an iterated toric fiber product of linear ideals. We also classify which Lawrence lifts of 2-way quasi-independence models are cTFPs and give a necessary condition under which a $k$-way model has ML-degree 1 using its facial submodels.

翻译：本文研究一类称为拟独立模型的对数线性统计模型的几何结构。环面纤维积对于理解这些模型中参数推断的几何性质非常有用，因为最大似然度在环面纤维积下具有可乘性。我们定义了坐标环面纤维积（cTFP），并给出了拟独立模型作为低阶模型的cTFP的充分必要条件。我们证明了所有最大似然度为1的二维拟独立模型的消失理想，均可实现为线性理想的迭代环面纤维积。我们还分类了哪些二维拟独立模型的劳伦斯提升是cTFP，并利用其面状子模型给出了$k$维模型具有最大似然度1的必要条件。