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MM algorithms for generalized BradleyTerry models
 The Annals of Statistics
, 2004
"... The Bradley–Terry model for paired comparisons is a simple and muchstudied means to describe the probabilities of the possible outcomes when individuals are judged against one another in pairs. Among the many studies of the model in the past 75 years, numerous authors have generalized it in several ..."
Abstract

Cited by 29 (1 self)
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The Bradley–Terry model for paired comparisons is a simple and muchstudied means to describe the probabilities of the possible outcomes when individuals are judged against one another in pairs. Among the many studies of the model in the past 75 years, numerous authors have generalized it in several directions, sometimes providing iterative algorithms for obtaining maximum likelihood estimates for the generalizations. Building on a theory of algorithms known by the initials MM, for minorization–maximization, this paper presents a powerful technique for producing iterative maximum likelihood estimation algorithms for a wide class of generalizations of the Bradley–Terry model. While algorithms for problems of this type have tended to be custombuilt in the literature, the techniques in this paper enable their mass production. Simple conditions are stated that guarantee that each algorithm described will produce a sequence that converges to the unique maximum likelihood estimator. Several of the algorithms and convergence results herein are new. 1. Introduction. In
Generalized bradleyterry models and multiclass probability estimates
 Journal of Machine Learning Research
"... Editor: The BradleyTerry model for obtaining individual skill from paired comparisons has been popular in many areas. In machine learning, this model is related to multiclass probability estimates by coupling all pairwise classification results. Error correcting output codes (ECOC) are a general f ..."
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Cited by 26 (3 self)
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Editor: The BradleyTerry model for obtaining individual skill from paired comparisons has been popular in many areas. In machine learning, this model is related to multiclass probability estimates by coupling all pairwise classification results. Error correcting output codes (ECOC) are a general framework to decompose a multiclass problem to several binary problems. To obtain probability estimates under this framework, this paper introduces a generalized BradleyTerry model in which paired individual comparisons are extended to paired team comparisons. We propose a simple algorithm with convergence proofs to solve the model and obtain individual skill. Experiments on synthetic and real data demonstrate that the algorithm is useful for obtaining multiclass probability estimates. Moreover, we discuss four extensions of the proposed model: 1) weighted individual skill, 2) homefield advantage, 3) ties, and 4) comparisons with more than two teams. Keywords: BradleyTerry model, Probability estimates, Error correcting output codes, Support Vector Machines
A BradleyTerry Artificial Neural Network Model for Individual Ratings in Group Competitions
, 2006
"... A common statistical model for paired comparisons is the BradleyTerry model. This research reparameterizes the BradleyTerry model as a singlelayer artificial neural network (ANN) and shows how it can be fitted using the delta rule. The ANN model is appealing because it makes using and extending ..."
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Cited by 4 (0 self)
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A common statistical model for paired comparisons is the BradleyTerry model. This research reparameterizes the BradleyTerry model as a singlelayer artificial neural network (ANN) and shows how it can be fitted using the delta rule. The ANN model is appealing because it makes using and extending the BradleyTerry model accessible to a broader community. It also leads to natural incremental and iterative updating methods. Several extensions are presented that allow the ANN model to learn to predict the outcome of complex, uneven twoteam group competitions by rating individual players—no other published model currently does this. An incrementallearning BradleyTerry ANN yields a probability estimate within less than 5 % of the actual value training over 3,379 multiplayer online matches of a popular teamand objectivebased firstperson shooter. Keywords: BradleyTerry model, paired comparisons, neural networks, delta rule, probability estimates
An Extension of Zermelo's Model for Ranking by Paired Comparisons
, 1999
"... this paper, we analyze a natural extension of Zermelo's model resulting from a singular perturbation. We show that this extension produces a ranking for arbitrary (nonnegative) outcome matrices and retains several of the desirable properties of the original model. In addition, we discuss computation ..."
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Cited by 3 (0 self)
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this paper, we analyze a natural extension of Zermelo's model resulting from a singular perturbation. We show that this extension produces a ranking for arbitrary (nonnegative) outcome matrices and retains several of the desirable properties of the original model. In addition, we discuss computational techniques and provide examples of their use. 1 Introduction
unknown title
, 2000
"... The existence of maximum likelihood estimates in the BradleyTerry model and its extensions ..."
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The existence of maximum likelihood estimates in the BradleyTerry model and its extensions
Statistical Analysis of the TM–Model via Bayesian Approach
"... The method of paired comparisons calls for the comparison of treatments presented in pairs to judges who prefer to better one based on their sensory evaluations. Thurstone (1927) and Mosteller (1951) em ploy the method of maximum likelihood to estimate the parameters of the ThurstoneMosteller model ..."
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The method of paired comparisons calls for the comparison of treatments presented in pairs to judges who prefer to better one based on their sensory evaluations. Thurstone (1927) and Mosteller (1951) em ploy the method of maximum likelihood to estimate the parameters of the ThurstoneMosteller model for the paired comparisons. A Bayesian analysis of the said model using the noninformative reference (Jeffreys) prior is presented in this study. The posterior estimates (means and joint modes) of the parameters and the posterior probabilities comparing the two parameters are obtained for the analysis. The predictive probabilities that one treatment (Ti) is preferred to any other treatment (Tj) in a future single comparison are also computed. In addition, the graphs of the marginal posterior distributions of the individual parameter are drawn. The appropriateness of the model is also tested using the different teststatistics.