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Empirical Similarity

In: Case-Based Predictions An Axiomatic Approach to Prediction, Classification and Statistical Learning

Author

Listed:
  • Itzhak Gilboa
  • Offer Lieberman
  • David Schmeidler

Abstract

An agent is asked to assess a real-valued variable Yp based on certain characteristics Xp=(Xp1,…,Xpm), and on a database consisting of (Xi1,…,Xim,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, Y¯ps, 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 Xp1,…,Xpm, associated with Yp, and the previously observed vector, Xi1,…,Xim. 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.

Suggested Citation

  • Itzhak Gilboa & Offer Lieberman & David Schmeidler, 2012. "Empirical Similarity," World Scientific Book Chapters, in: Case-Based Predictions An Axiomatic Approach to Prediction, Classification and Statistical Learning, chapter 9, pages 211-243, World Scientific Publishing Co. Pte. Ltd..
  • Handle: RePEc:wsi:wschap:9789814366182_0009
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    References listed on IDEAS

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    1. Antoine Billot & Itzhak Gilboa & David Schmeidler, 2012. "Axiomatization of an Exponential Similarity Function," World Scientific Book Chapters, in: Case-Based Predictions An Axiomatic Approach to Prediction, Classification and Statistical Learning, chapter 10, pages 245-257, World Scientific Publishing Co. Pte. Ltd..
    2. Itzhak Gilboa & David Schmeidler, 1995. "Case-Based Decision Theory," The Quarterly Journal of Economics, President and Fellows of Harvard College, 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.
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    More about this item

    Keywords

    Case-Based Reasoning; Case-Based Decision Theory; Statistics; Decision Under Uncertainty;
    All these keywords.

    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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