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Register a new userPrediction and Evaluation in College Hockey using the Bradley-Terry-Zermelo Model
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Added by
John T. Whelan
Publication date
2020
fields
Mathematical Statistics
and research's language is
English
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We describe the application of the Bradley-Terry model to NCAA Division I Mens 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.
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Identifying the most deprived regions of any country or city is key if policy makers are to design successful interventions. However, locating areas with the greatest need is often surprisingly challenging in developing countries. Due to the logistical challenges of traditional household surveying, official statistics can be slow to be updated; estimates that exist can be coarse, a consequence of prohibitive costs and poor infrastructures; and mass urbanisation can render manually surveyed figures rapidly out-of-date. Comparative judgement models, such as the Bradley--Terry model, offer a promising solution. Leveraging local knowledge, elicited via comparisons of different areas affluence, such models can both simplify logistics and circumvent biases inherent to house-hold surveys. Yet widespread adoption remains limited, due to the large amount of data existing approaches still require. We address this via development of a novel Bayesian Spatial Bradley--Terry model, which substantially decreases the amount of data comparisons required for effective inference. This model integrates a network representation of the city or country, along with assumptions of spatial smoothness that allow deprivation in one area to be informed by neighbouring areas. We demonstrate the practical effectiveness of this method, through a novel comparative judgement data set collected in Dar es Salaam, Tanzania.
A common problem faced in statistical inference is drawing conclusions from paired comparisons, in which two objects compete and one is declared the victor. A probabilistic approach to such a problem is the Bradley-Terry model, first studied by Zermelo in 1929 and rediscovered by Bradley and Terry in 1952. One obvious area of application for such a model is sporting events, and in particular Major League Baseball. With this in mind, we describe a hierarchical Bayesian version of Bradley-Terry suitable for use in ranking and prediction problems, and compare results from these application domains to standard maximum likelihood approaches. Our Bayesian methods outperform the MLE-based analogues, while being simple to construct, implement, and interpret.
We propose a time-varying generalization of the Bradley-Terry model that allows for nonparametric modeling of dynamic global rankings of distinct teams. We develop a novel estimator that relies on kernel smoothing to pre-process the pairwise comparisons over time and is applicable in sparse settings where the Bradley-Terry may not be fit. We obtain necessary and sufficient conditions for the existence and uniqueness of our estimator. We also derive time-varying oracle bounds for both the estimation error and the excess risk in the model-agnostic setting where the Bradley-Terry model is not necessarily the true data generating process. We thoroughly test the practical effectiveness of our model using both simulated and real world data and suggest an efficient data-driven approach for bandwidth tuning.
The Bradley-Terry model assigns probabilities for the outcome of paired comparison experiments based on strength parameters associated with the objects being compared. We consider different proposed choices of prior parameter distributions for Bayesian inference of the strength parameters based on the paired comparison results. We evaluate them according to four desiderata motivated by the use of inferred Bradley-Terry parameters to rate teams on the basis of outcomes of a set of games: invariance under interchange of teams, invariance under interchange of winning and losing, normalizability and invariance under elimination of teams. We consider various proposals which fail to satisfy one or more of these desiderata, and illustrate two proposals which satisfy them. Both are one-parameter independent distributions for the logarithms of the team strengths: 1) Gaussian and 2) Type III generalized logistic.
Optimal design theory for nonlinear regression studies local optimality on a given design space. We identify designs for the Bradley--Terry paired comparison model with small undirected graphs and prove that every saturated D-optimal design is represented by a path. We discuss the case of four alternatives in detail and derive explicit polynomial inequality descriptions for optimality regions in parameter space. Using these regions, for each point in parameter space we can prescribe a D-optimal design.
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