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Empirical Similarity


  • Itzhak Gilboa

    () (GREGH - Groupement de Recherche et d'Etudes en Gestion à HEC - HEC Paris - Ecole des Hautes Etudes Commerciales - CNRS - Centre National de la Recherche Scientifique, Tel Aviv University [Tel Aviv])

  • David Schmeidler

    (Tel Aviv University [Tel Aviv])

  • Offer Lieberman


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.

Suggested Citation

  • Itzhak Gilboa & David Schmeidler & Offer Lieberman, 2006. "Empirical Similarity," Post-Print hal-00746558, HAL.
  • Handle: RePEc:hal:journl:hal-00746558
    DOI: 10.1162/rest.88.3.433
    Note: View the original document on HAL open archive server:

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    References listed on IDEAS

    1. Billot, Antoine & Gilboa, Itzhak & Schmeidler, David, 2008. "Axiomatization of an exponential similarity function," Mathematical Social Sciences, Elsevier, vol. 55(2), pages 107-115, March.
    2. Itzhak Gilboa & David Schmeidler, 1995. "Case-Based Decision Theory," The Quarterly Journal of Economics, Oxford University Press, vol. 110(3), pages 605-639.
    3. Gayer Gabrielle & Gilboa Itzhak & Lieberman Offer, 2007. "Rule-Based and Case-Based Reasoning in Housing Prices," The B.E. Journal of Theoretical Economics, De Gruyter, vol. 7(1), pages 1-37, April.
    4. Antoine Billot & Itzhak Gilboa & Dov Samet & David Schmeidler, 2003. "Probabilities: Frequencies Viewed in Perspective," Levine's Bibliography 666156000000000295, UCLA Department of Economics.
    5. Antoine Billot & Itzhak Gilboa & David Schmeidler, 2004. "An Axiomatization of an Exponential Similarity Function," Levine's Bibliography 122247000000000678, UCLA Department of Economics.
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    More about this item


    Similarity; estimation;

    JEL classification:

    • C1 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General
    • C8 - Mathematical and Quantitative Methods - - Data Collection and Data Estimation Methodology; Computer Programs
    • D8 - Microeconomics - - Information, Knowledge, and Uncertainty


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