The standard mathematical approach to fourth-down decision making in American football is to make the decision that maximizes estimated win probability. Win probability estimates arise from machine learning models fit from historical data. These models attempt to capture a nuanced relationship between a noisy binary outcome variable and game-state variables replete with interactions and non-linearities from a finite dataset of just a few thousand games. Thus, it is imperative to knit uncertainty quantification into the fourth-down decision procedure; we do so using bootstrapping. We find that uncertainty in the estimated optimal fourth-down decision is far greater than that currently expressed by sports analysts in popular sports media.

翻译：美式橄榄球四档决策的标准数学方法是选择能够最大化估计获胜概率的决策。获胜概率估计源自基于历史数据拟合的机器学习模型。这些模型试图从仅包含数千场比赛的有限数据集中，捕捉一个存在交互效应和非线性关系的比赛状态变量与一个具有噪声的二元结果变量之间的微妙关系。因此，必须将不确定性量化纳入四档决策过程；我们通过自助法实现了这一点。我们发现，当前体育媒体中体育分析师所表达的决策不确定性，远低于对最优四档决策估计本身所包含的不确定性。