Recovering weights from comparison results in extensions of BTL model

Details

Supervisors

Hendrickx, Julien

Faculty

Ecole polytechnique de Louvain

Degree label

Master [120] : ingénieur civil en mathématiques appliquées, à finalité spécialisée

Abstract

In this master thesis, we generalize the weighted least squares method developed for the BTL model to models extending the latter one. This is done by extending the Gauss-Markov theorem, which finds the Best Linear Unbiased Estimator of parameters from a linear system with noisy observations. This problem of recovering the parameters of a probability model aims to estimate the quality of the items used for example for user services (streaming platform, e-commerce), sport tournament, sociology, and many others. Observations of such scores are often non-existent or poorly calibrated, as opposed to preference observations. Several preference models based on item scores exist, such as the BTL model, based on a probability wi/(wi +wj) that an item i wins over an item j, and various extensions of this model. Methods have been developed to estimate the vector of scores w = (w1, · · · , wN ) based on BTL model. This master thesis aims to generalize the method developed in [HOS20] for BTL to recover item scores of BTL-extended models for comparisons depending on those scores and possibly on additional parameters. This thesis focuses on two pairwise comparisons models allowing ties in the observations and on two triple comparisons models without ties. The extended weighted least squares method is designed with a focus on the weight matrix and has been justified by theoretical results in the context of linear estimators. For this purpose, Gauss-Markov Theorem for BLUE estimators of observable parameters has been generalized for minimum error estimator with possibly unobservable parameters and non-independent observations. Furthermore, we characterize the decrease of the error and we observe that this method outperforms the least squares method and iterative maximum likelihood for the different BTL-extended models.