  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.
