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Stan Lipovetsky - One of the best experts on this subject based on the ideXlab platform.
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priorities in Thurstone scaling and steady state probabilities in markov stochastic modeling
Journal of Modern Applied Statistical Methods, 2013Co-Authors: Stan LipovetskyAbstract:Thurstone scaling is widely used in marketing and advertising research where various methods of applied psychology are utilized. This article considers several analytical tools useful for positioning a set of items on a Thurstone scale via regression modeling and Markov stochastic processing in the form of ChapmanKolmogorov equations. These approaches produce interval and ratio scales of preferences and enrich the possibilities of paired comparison estimation applied for solving practical problems of prioritization and probability of choice modeling.
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priority and choice probability estimation by ranking rating and combined data
Journal of statistical theory and practice, 2007Co-Authors: Stan LipovetskyAbstract:Ranking data is commonly used in marketing and advertising research for priority estimation among the compared items by Thurstone scaling. Rating data is also often used in TURF, or total unduplicated reach and frequency analysis to find the best items. Both ranks and rates data sets can be elicited and utilized simultaneously to obtain a combined preference estimation. This work develops several techniques of priority evaluation. It considers maximum likelihood of the order statistics for the ranking data with the probit, logit, and multinomial links for the Thurstone scale. Non-linear optimization with the least squares or maximum likelihood objective is introduced for TURF modeling. Combined estimation by both rank and rate data is suggested in singular value decomposition and Geary-Khamis equation approaches. The proposed methods produce priorities among the compared items and probabilities of their choice.
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Thurstone scaling in order statistics
Mathematical and Computer Modelling, 2007Co-Authors: Stan LipovetskyAbstract:Thurstone scaling is widely used for presenting the priorities among the compared items. The mean values of the quantiles corresponding to frequencies of each stimilus' preference over the other stimuli define the items' locations on the psychological continuum of the Thurstone scale. This paper considers an extension of the scale levels to the aggregates of the independent covariates. In a sense, it is similar to a multiple regression extension of the mean value of the dependent variable to its conditional mean expressed by the linear aggregate of the independent variables. A maximum likelihood objective constructed by the probabilities of the order statistics applied to the ranked or paired comparison data is suggested. Probit, logit and multinomial links are tried to obtain the Thurstonian scale exposition by the covariates, and to estimate probabilities of the items' choice. This approach is very convenient and can substantially enrich both theoretical interpretation and practical application of Thurstone modelling, particularly in marketing and advertising research.
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Thurstone scaling via binary response regression
Statistical Methodology, 2004Co-Authors: Stan Lipovetsky, Michael W ConklinAbstract:Abstract Thurstone scaling is a widely used tool in marketing research, as well as in areas of applied psychology. The positions of the compared items, or stimuli on a Thurstone scale are estimated by averaging the quantiles corresponding to frequencies of each stimulus’s preference over the other stimuli. We consider maximum likelihood estimation for Thurstone scaling that utilizes paired comparison data. From this perspective we obtain a binary response regression with a probit or logit link. In addition to the levels on a psychological scale, the suggested approach produces standard errors, t -statistics, and other characteristics of regression quality. This approach can help in both the theoretical interpretation and the practical application of Thurstone modeling.
Martin J Wainwright - One of the best experts on this subject based on the ideXlab platform.
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stochastically transitive models for pairwise comparisons statistical and computational issues
IEEE Transactions on Information Theory, 2017Co-Authors: Nihar B Shah, Sivaraman Balakrishnan, Adityanand Guntuboyina, Martin J WainwrightAbstract:There are various parametric models for analyzing pairwise comparison data, including the Bradley–Terry–Luce (BTL) and Thurstone models, but their reliance on strong parametric assumptions is limiting. In this paper, we study a flexible model for pairwise comparisons, under which the probabilities of outcomes are required only to satisfy a natural form of stochastic transitivity. This class includes parametric models, including the BTL and Thurstone models as special cases, but is considerably more general. We provide various examples of models in this broader stochastically transitive class for which classical parametric models provide poor fits. Despite this greater flexibility, we show that the matrix of probabilities can be estimated at the same rate as in standard parametric models up to logarithmic terms. On the other hand, unlike in the BTL and Thurstone models, computing the minimax-optimal estimator in the stochastically transitive model is non-trivial, and we explore various computationally tractable alternatives. We show that a simple singular value thresholding algorithm is statistically consistent but does not achieve the minimax rate. We then propose and study algorithms that achieve the minimax rate over interesting sub-classes of the full stochastically transitive class. We complement our theoretical results with thorough numerical simulations.
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stochastically transitive models for pairwise comparisons statistical and computational issues
International Conference on Machine Learning, 2016Co-Authors: Nihar B Shah, Sivaraman Balakrishnan, Adityanand Guntuboyina, Martin J WainwrightAbstract:There are various parametric models for analyzing pairwise comparison data, including the Bradley-Terry-Luce (BTL) and Thurstone models, but their reliance on strong parametric assumptions is limiting. In this work, we study a flexible model for pairwise comparisons, under which the probabilities of outcomes are required only to satisfy a natural form of stochastic transitivity. This class includes parametric models including the BTL and Thurstone models as special cases, but is considerably more general. We provide various examples of models in this broader stochastically transitive class for which classical parametric models provide poor fits. Despite this greater flexibility, we show that the matrix of probabilities can be estimated at the same rate as in standard parametric models. On the other hand, unlike in the BTL and Thurstone models, computing the minimax-optimal estimator in the stochastically transitive model is non-trivial, and we explore various computationally tractable alternatives. We show that a simple singular value thresholding algorithm is statistically consistent but does not achieve the minimax rate. We then propose and study algorithms that achieve the minimax rate over interesting sub-classes of the full stochastically transitive class. We complement our theoretical results with thorough numerical simulations.
Se-young Yun - One of the best experts on this subject based on the ideXlab platform.
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Parameter Estimation for Thurstone Choice Models
arXiv: Statistics Theory, 2017Co-Authors: Milan Vojnovic, Se-young YunAbstract:We consider the estimation accuracy of individual strength parameters of a Thurstone choice model when each input observation consists of a choice of one item from a set of two or more items (so called top-1 lists). This model accommodates the well-known choice models such as the Luce choice model for comparison sets of two or more items and the Bradley-Terry model for pair comparisons. We provide a tight characterization of the mean squared error of the maximum likelihood parameter estimator. We also provide similar characterizations for parameter estimators defined by a rank-breaking method, which amounts to deducing one or more pair comparisons from a comparison of two or more items, assuming independence of these pair comparisons, and maximizing a likelihood function derived under these assumptions. We also consider a related binary classification problem where each individual parameter takes value from a set of two possible values and the goal is to correctly classify all items within a prescribed classification error.
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parameter estimation for generalized Thurstone choice models
International Conference on Machine Learning, 2016Co-Authors: Milan Vojnovic, Se-young YunAbstract:We consider the maximum likelihood parameter estimation problem for a generalized Thurstone choice model, where choices are top-1 items from comparison sets of two or more items. We provide tight characterizations of the mean square error, as well as necessary and sufficient conditions for correct classification when each item belongs to one of two classes. These results provide insights into how the estimation accuracy depends on the choice of a generalized Thurstone choice model and the structure of comparison sets. We find that for a priori unbiased structures of comparisons, e.g., when comparison sets are drawn independently and uniformly at random, the number of observations needed to achieve a prescribed estimation accuracy depends on the choice of a generalized Thurstone choice model. For a broad set of generalized Thurstone choice models, which includes all popular instances used in practice, the estimation error is shown to be largely insensitive to the cardinality of comparison sets. On the other hand, we found that there exist generalized Thurstone choice models for which the estimation error decreases much faster with the cardinality of comparison sets.
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Parameter estimation for generalized Thurstone choice models
LSE Research Online Documents on Economics, 2016Co-Authors: Milan Vojnovic, Se-young YunAbstract:We consider the maximum likelihood parameter estimation problem for a generalized Thurstone choice model, where choices are from comparison sets of two or more items. We provide tight characterizations of the mean square error, as well as necessary and sufficient conditions for correct classification when each item belongs to one of two classes. These results provide insights into how the estimation accuracy depends on the choice of a generalized Thurstone choice model and the structure of comparison sets. We find that for a priori unbiased structures of comparisons, e.g., when comparison sets are drawn independently and uniformly at random, the number of observations needed to achieve a prescribed estimation accuracy depends on the choice of a generalized Thurstone choice model. For a broad set of generalized Thurstone choice models, which includes all popular instances used in practice, the estimation error is shown to be largely insensitive to the cardinality of comparison sets. On the other hand, we found that there exist generalized Thurstone choice models for which the estimation error decreases much faster with the cardinality of comparison sets.
Nihar B Shah - One of the best experts on this subject based on the ideXlab platform.
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stochastically transitive models for pairwise comparisons statistical and computational issues
IEEE Transactions on Information Theory, 2017Co-Authors: Nihar B Shah, Sivaraman Balakrishnan, Adityanand Guntuboyina, Martin J WainwrightAbstract:There are various parametric models for analyzing pairwise comparison data, including the Bradley–Terry–Luce (BTL) and Thurstone models, but their reliance on strong parametric assumptions is limiting. In this paper, we study a flexible model for pairwise comparisons, under which the probabilities of outcomes are required only to satisfy a natural form of stochastic transitivity. This class includes parametric models, including the BTL and Thurstone models as special cases, but is considerably more general. We provide various examples of models in this broader stochastically transitive class for which classical parametric models provide poor fits. Despite this greater flexibility, we show that the matrix of probabilities can be estimated at the same rate as in standard parametric models up to logarithmic terms. On the other hand, unlike in the BTL and Thurstone models, computing the minimax-optimal estimator in the stochastically transitive model is non-trivial, and we explore various computationally tractable alternatives. We show that a simple singular value thresholding algorithm is statistically consistent but does not achieve the minimax rate. We then propose and study algorithms that achieve the minimax rate over interesting sub-classes of the full stochastically transitive class. We complement our theoretical results with thorough numerical simulations.
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stochastically transitive models for pairwise comparisons statistical and computational issues
International Conference on Machine Learning, 2016Co-Authors: Nihar B Shah, Sivaraman Balakrishnan, Adityanand Guntuboyina, Martin J WainwrightAbstract:There are various parametric models for analyzing pairwise comparison data, including the Bradley-Terry-Luce (BTL) and Thurstone models, but their reliance on strong parametric assumptions is limiting. In this work, we study a flexible model for pairwise comparisons, under which the probabilities of outcomes are required only to satisfy a natural form of stochastic transitivity. This class includes parametric models including the BTL and Thurstone models as special cases, but is considerably more general. We provide various examples of models in this broader stochastically transitive class for which classical parametric models provide poor fits. Despite this greater flexibility, we show that the matrix of probabilities can be estimated at the same rate as in standard parametric models. On the other hand, unlike in the BTL and Thurstone models, computing the minimax-optimal estimator in the stochastically transitive model is non-trivial, and we explore various computationally tractable alternatives. We show that a simple singular value thresholding algorithm is statistically consistent but does not achieve the minimax rate. We then propose and study algorithms that achieve the minimax rate over interesting sub-classes of the full stochastically transitive class. We complement our theoretical results with thorough numerical simulations.
Milan Vojnovic - One of the best experts on this subject based on the ideXlab platform.
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Parameter Estimation for Thurstone Choice Models
arXiv: Statistics Theory, 2017Co-Authors: Milan Vojnovic, Se-young YunAbstract:We consider the estimation accuracy of individual strength parameters of a Thurstone choice model when each input observation consists of a choice of one item from a set of two or more items (so called top-1 lists). This model accommodates the well-known choice models such as the Luce choice model for comparison sets of two or more items and the Bradley-Terry model for pair comparisons. We provide a tight characterization of the mean squared error of the maximum likelihood parameter estimator. We also provide similar characterizations for parameter estimators defined by a rank-breaking method, which amounts to deducing one or more pair comparisons from a comparison of two or more items, assuming independence of these pair comparisons, and maximizing a likelihood function derived under these assumptions. We also consider a related binary classification problem where each individual parameter takes value from a set of two possible values and the goal is to correctly classify all items within a prescribed classification error.
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parameter estimation for generalized Thurstone choice models
International Conference on Machine Learning, 2016Co-Authors: Milan Vojnovic, Se-young YunAbstract:We consider the maximum likelihood parameter estimation problem for a generalized Thurstone choice model, where choices are top-1 items from comparison sets of two or more items. We provide tight characterizations of the mean square error, as well as necessary and sufficient conditions for correct classification when each item belongs to one of two classes. These results provide insights into how the estimation accuracy depends on the choice of a generalized Thurstone choice model and the structure of comparison sets. We find that for a priori unbiased structures of comparisons, e.g., when comparison sets are drawn independently and uniformly at random, the number of observations needed to achieve a prescribed estimation accuracy depends on the choice of a generalized Thurstone choice model. For a broad set of generalized Thurstone choice models, which includes all popular instances used in practice, the estimation error is shown to be largely insensitive to the cardinality of comparison sets. On the other hand, we found that there exist generalized Thurstone choice models for which the estimation error decreases much faster with the cardinality of comparison sets.
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Parameter estimation for generalized Thurstone choice models
LSE Research Online Documents on Economics, 2016Co-Authors: Milan Vojnovic, Se-young YunAbstract:We consider the maximum likelihood parameter estimation problem for a generalized Thurstone choice model, where choices are from comparison sets of two or more items. We provide tight characterizations of the mean square error, as well as necessary and sufficient conditions for correct classification when each item belongs to one of two classes. These results provide insights into how the estimation accuracy depends on the choice of a generalized Thurstone choice model and the structure of comparison sets. We find that for a priori unbiased structures of comparisons, e.g., when comparison sets are drawn independently and uniformly at random, the number of observations needed to achieve a prescribed estimation accuracy depends on the choice of a generalized Thurstone choice model. For a broad set of generalized Thurstone choice models, which includes all popular instances used in practice, the estimation error is shown to be largely insensitive to the cardinality of comparison sets. On the other hand, we found that there exist generalized Thurstone choice models for which the estimation error decreases much faster with the cardinality of comparison sets.
