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Prediction and modelling of competitive sports outcomes has received much recent attention, especially from the Bayesian statistics and machine learning communities. In the real world setting of outcome prediction, the seminal {E}lH{o} update still remains, after more than 50 years, a valuable baseline which is difficult to improve upon, though in its original form it is a heuristic and not a proper statistical model. Mathematically, the {E}lH{o} rating system is very closely related to the Bradley-Terry models, which are usually used in an explanatory fashion rather than in a predictive supervised or on-line learning setting. Exploiting this close link between these two model classes and some newly observed similarities, we propose a new supervised learning framework with close similarities to logistic regression, low-rank matrix completion and neural networks. Building on it, we formulate a class of structured log-odds models, unifying the desirable properties found in the above: supervised probabilistic prediction of scores and wins/draws/losses, batch/epoch and on-line learning, as well as the possibility to incorporate features in the prediction, without having to sacrifice simplicity, parsimony of the Bradley-Terry models, or computational efficiency of {E}lH{o}s original approach. We validate the structured log-odds modelling approach in synthetic experiments and English Premier League outcomes, where the added expressivity yields the best predictions reported in the state-of-art, close to the quality of contemporary betting odds.
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 Bayesi
Recently proposed consistency-based Semi-Supervised Learning (SSL) methods such as the $Pi$-model, temporal ensembling, the mean teacher, or the virtual adversarial training, have advanced the state of the art in several SSL tasks. These methods can
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 comparis
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 Zerme
In many application settings, the data have missing entries which make analysis challenging. An abundant literature addresses missing values in an inferential framework: estimating parameters and their variance from incomplete tables. Here, we consid