Completion problems, of recovering a point from a set of observed coordinates, are abundant in applications to image reconstruction, phylogenetics, and data science. We consider a completion problem coming from algebraic statistics: to describe the completions of a point to a probability distribution lying in a given log-linear model. When there are finitely many completions, we show that these points either have a unique completion or two completions to the log-linear model depending on the set of observed coordinates. We additionally describe the region of points which have a completion to the log-linear model.

翻译：补全问题——即从观测到的坐标集合中恢复一个点——在图像重建、系统发育学和数据科学等应用中广泛存在。本文考虑一个源于代数统计的补全问题：描述一个点补全为位于给定对数线性模型中的概率分布的所有可能方式。当补全结果有限时，我们证明这些点要么具有唯一补全，要么根据观测坐标集合的不同，存在两种补全到对数线性模型的方式。此外，我们还刻画了能够补全到该对数线性模型的点所构成的区域。