Abstract

We investigate the geometry of a family of log-linear statistical models called quasi-independence models. The toric fiber product (TFP) 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. This implies that all such models have a parametrization under which the iterative proportional scaling procedure computes the exact symbolic maximum likelihood estimate in one iteration. 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.

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