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Empirical Similarity

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  • Itzhak Gilboa
  • Offer Lieberman
  • David Schmeidler

Abstract

An agent is asked to assess a real-valued variable Y_{p} based on certain characteristics X_{p} = (X_{p}^{1},...,X_{p}^{m}), and on a database consisting (X_{i}^{1},...,X_{i}^{m},Y_{i}) 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, Y_{p}^{s}, be the weighted average of all previously observed values Y_{i}, where the weight of Y_{i}, for every i =1,...,n, is the similarity between the vector X_{p}^{1},...,X_{p}^{m}, associated with Y_{p}, and the previously observed vector, X_{i}^{1},...,X_{i}^{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 functional 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.

(This abstract was borrowed from another version of this item.)

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File URL: http://www.tau.ac.il/~igilboa/pdf/GLS_Empirical_Similarity.pdf
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Bibliographic Info

Paper provided by UCLA Department of Economics in its series Levine's Bibliography with number 122247000000000684.

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Date of creation: 09 Nov 2004
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Handle: RePEc:cla:levrem:122247000000000684

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  1. Gilboa, Itzhak & Schmeidler, David, 1995. "Case-Based Decision Theory," The Quarterly Journal of Economics, MIT Press, MIT Press, vol. 110(3), pages 605-39, August.
  2. Gabrielle Gayer & Itzhak Gilboa & Offer Lieberman, 2004. "Rule-Based and Case-Based Reasoning in Housing Prices," Cowles Foundation Discussion Papers, Cowles Foundation for Research in Economics, Yale University 1493, Cowles Foundation for Research in Economics, Yale University.
  3. Antoine Billot & Itzhak Gilboa & David Schmeidler, 2004. "Axiomatization of an Exponential Similarity Function," Cowles Foundation Discussion Papers, Cowles Foundation for Research in Economics, Yale University 1485, Cowles Foundation for Research in Economics, Yale University.
  4. Antoine Billot & Itzhak Gilboa & David Schmeidler, 2004. "An Axiomatization of an Exponential Similarity Function," Levine's Bibliography 122247000000000678, UCLA Department of Economics.
  5. Antoine Billot & Itzhak Gilboa & Dov Samet & David Schmeidler, 2003. "Probabilities: Frequencies Viewed in Perspective," Levine's Bibliography 666156000000000295, UCLA Department of Economics.
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