The maximum likelihood threshold of a statistical model is the minimum number of datapoints required to fit the model via maximum likelihood estimation. In this paper we determine the maximum likelihood thresholds of generic linear concentration models. This turns out to be the number one would expect from a naive dimension count, which is surprising and nontrivial to prove given that the maximum likelihood threshold is a semi-algebraic concept. We also describe geometrically how a linear concentration model can fail to exhibit this generic behavior and briefly discuss connections to rigidity theory.

翻译：统计模型的最大似然阈值是指通过最大似然估计拟合该模型所需的最小数据点数。本文确定了通用线性浓度模型的最大似然阈值。该结果与通过朴素维度计数推测的数值一致，这一结论令人惊讶且证明颇具挑战性——因为最大似然阈值本质上属于半代数概念。我们还从几何角度描述了线性浓度模型未能展现这一通用特征的具体机制，并简要讨论了其与刚性理论之间的关联。