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Learning preference distributions is a key problem in many areas (e.g., recommender systems, IR, social choice). However, existing methods require restrictive data models for evidence about user preferences. We relax these restrictions by considering as data arbitrary pairwise comparisons---the fundamental building blocks of ordinal rankings. We develop the first algorithms for learning Mallows models (and mixtures) with pairwise comparisons. At the heart is a new algorithm, the generalized repeated insertion model (GRIM), for sampling from arbitrary ranking distributions. We develop approximate samplers that are exact for many important special cases---and have provable bounds with pairwise evidence---and derive algorithms for evaluating log-likelihood, learning Mallows mixtures, and non-parametric estimation. Experiments on large, real-world datasets show the effectiveness of our approach.
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