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Empirical Similarity

Abstract : An agent is asked to assess a real-valued variable Yp based on certain characteristics Xp = (Xp-super-1, ..., Xp-super-m), and on a database consisting of Xi-super-1, ... Xi-super-m, Yi) for i = 1, ..., n. A possible approach to combine past observations of X and Y with the current values of X to generate an assessment of Y is similarity-weighted averaging. It suggests that the predicted value of Y, Ȳp-super-s, be the weighted average of all previously observed values Yi, where the weight of Yi for every i = 1, ..., n, is the similarity between the vector Xp-super-1, ..., Xp-super-m, associated with Yp, and the previously observed vector, Xi-super-1, ..., Xi-super-m. We axiomatize this rule. We assume that, given every database, a predictor has a ranking over possible values, and we show that certain reasonable conditions on these rankings imply that they are determined by the proximity to a similarity-weighted average for a certain similarity function. The axiomatization does not suggest a particular similarity function, or even a particular form of this function. We therefore proceed to suggest that the similarity function be estimated from past observations.We develop tools of statistical inference for parametric estimation of the similarity function, for the case of a continuous as well as a discrete variable. Finally, we discuss the relationship of the proposed method to other methods of estimation and prediction. Copyright by the President and Fellows of Harvard College and the Massachusetts Institute of Technology.
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Contributor : Antoine Haldemann Connect in order to contact the contributor
Submitted on : Monday, October 29, 2012 - 1:02:00 PM
Last modification on : Thursday, January 11, 2018 - 6:19:31 AM





Itzhak Gilboa, David Schmeidler, Offer Lieberman. Empirical Similarity. Review of Economics and Statistics, Massachusetts Institute of Technology Press (MIT Press), 2006, vol. 88, issue 3, pp. 433-444. ⟨10.1162/rest.88.3.433⟩. ⟨hal-00746558⟩



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