We describe the application of the Bradley-Terry model to NCAA Division I Men's Ice Hockey. A Bayesian construction gives a joint posterior probability distribution for the log-strength parameters, given a set of game results and a choice of prior distribution. For several suitable choices of prior, it is straightforward to find the maximum a posteriori point (MAP) and a Hessian matrix, allowing a Gaussian approximation to be constructed. Posterior predictive probabilities can be estimated by 1) setting the log-strengths to their MAP values, 2) using the Gaussian approximation for analytical or Monte Carlo integration, or 3) applying importance sampling to re-weight the results of a Monte Carlo simulation. We define a method to evaluate any models which generate predicted probabilities for future outcomes, using the Bayes factor given the actual outcomes, and apply it to NCAA tournament results. Finally, we describe an on-line tool which currently estimates probabilities of future results using MAP evaluation and describe how it can be refined using the Gaussian approximation or importance sampling.

翻译：本文描述了Bradley-Terry模型在NCAA一级男子冰球比赛中的应用。通过贝叶斯构造，在给定一系列比赛结果和先验分布选择的情况下，可获得对数强度参数的联合后验概率分布。对于几种合适的先验分布选择，可以简便地求得最大后验点及其Hessian矩阵，从而构建高斯近似。后验预测概率的估计可通过以下三种方式实现：1)将对数强度设置为最大后验点值，2)利用高斯近似进行解析或蒙特卡洛积分，3)应用重要性采样对蒙特卡洛模拟结果进行加权。我们提出了一种评估方法，该方法基于实际结果的贝叶斯因子，用于评估任何对未来结果生成预测概率的模型，并将其应用于NCAA锦标赛结果。最后，我们描述了一个当前利用最大后验点估计评估未来结果概率的在线工具，并说明了如何通过高斯近似或重要性采样对该工具进行改进。