We import the algebro-geometric notion of a complete collineation into the study of maximum likelihood estimation in directed Gaussian graphical models. A complete collineation produces a perturbation of sample data, which we call a stabilisation of the sample. While a maximum likelihood estimate (MLE) may not exist or be unique given sample data, it is always unique given a stabilisation. We relate the MLE given a stabilisation to the MLE given original sample data, when one exists, providing necessary and sufficient conditions for the MLE given a stabilisation to be one given the original sample. For linear regression models, we show that the MLE given any stabilisation is the minimal norm choice among the MLEs given an original sample. We show that the MLE has a well-defined limit as the stabilisation of a sample tends to the original sample, and that the limit is an MLE given the original sample, when one exists. Finally, we study which MLEs given a sample can arise as such limits. We reduce this to a question regarding the non-emptiness of certain algebraic varieties.

翻译：我们将代数几何中的完整共线关系概念引入有向高斯图模型的最大似然估计研究中。完整共线关系产生样本数据的扰动，我们称之为样本的稳定性化处理。尽管给定样本数据时最大似然估计可能不存在或不唯一，但在给定稳定性化处理后它总是唯一的。我们将给定稳定性化处理后的最大似然估计与原始样本数据下的最大似然估计（当存在时）建立联系，给出了给定稳定性化处理的估计为原始样本下估计的充要条件。对于线性回归模型，我们证明任何稳定性化处理后的最大似然估计都是原始样本下所有估计中具有最小范数的选择。我们证明当样本的稳定性化处理趋于原始样本时，最大似然估计具有确定的极限，且该极限是原始样本下的最大似然估计（当存在时）。最后，我们研究哪些样本下的最大似然估计可以作为此类极限出现，并将该问题归结为特定代数簇非空性的判定。