Prediction and evaluation in College Hockey using the Bradley–Terry–Zermelo model

Abstract

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