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Axiomatization of an Exponential Similarity Function

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

Author

Listed:
  • Antoine Billot
  • Itzhak Gilboa
  • David Schmeidler

Abstract

An individual is asked to assess a real-valued variable y based on certain characteristics x = (x1, …, xm), and on a database consisting of n observations of (x1, …, xm, y). 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, yn+1s, be the weighted average of all previously observed values yi, where the weight of yi is the similarity between the vector xn+11,…,xn+1m, associated with yn+1, and the previously observed vector, xi1,…,xim. This paper axiomatizes, in terms of the prediction yn+1, a similarity function that is a (decreasing) exponential in a norm of the difference between the two vectors compared.

Suggested Citation

  • 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..
  • Handle: RePEc:wsi:wschap:9789814366182_0010
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    as
    1. Itzhak Gilboa & David Schmeidler, 2003. "Inductive Inference: An Axiomatic Approach," Econometrica, Econometric Society, vol. 71(1), pages 1-26, January.
    2. Antoine Billot & Itzhak Gilboa & Dov Samet & David Schmeidler, 2012. "Probabilities as Similarity-Weighted Frequencies," World Scientific Book Chapters, in: Case-Based Predictions An Axiomatic Approach to Prediction, Classification and Statistical Learning, chapter 7, pages 169-184, World Scientific Publishing Co. Pte. Ltd..
    3. 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..
    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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    Citations

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    Cited by:

    1. Todd Guilfoos & Andreas Duus Pape, 2020. "Estimating Case-Based Learning," Games, MDPI, vol. 11(3), pages 1-25, September.
    2. Antoine Billot & Itzhak Gilboa & Dov Samet & David Schmeidler, 2012. "Probabilities as Similarity-Weighted Frequencies," World Scientific Book Chapters, in: Case-Based Predictions An Axiomatic Approach to Prediction, Classification and Statistical Learning, chapter 7, pages 169-184, World Scientific Publishing Co. Pte. Ltd..
    3. 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..
    4. Huirong Zhang & Zhenyu Zhang & Lixin Zhou & Shuangsheng Wu, 2021. "Case-Based Reasoning for Hidden Property Analysis of Judgment Debtors," Mathematics, MDPI, vol. 9(13), pages 1-17, July.
    5. Wolfgang Ossadnik & Dirk Wilmsmann & Benedikt Niemann, 2013. "Experimental evidence on case-based decision theory," Theory and Decision, Springer, vol. 75(2), pages 211-232, August.
    6. Bleile, Jörg, 2016. "Cautious Belief Formation," Center for Mathematical Economics Working Papers 507, Center for Mathematical Economics, Bielefeld University.
    7. Guerdjikova, Ani, 2004. "Preference for diversification with similarity considerations," Papers 04-48, Sonderforschungsbreich 504.
    8. Gilboa, Itzhak & Lieberman, Offer & Schmeidler, David, 2011. "A similarity-based approach to prediction," Journal of Econometrics, Elsevier, vol. 162(1), pages 124-131, May.
    9. Golosnoy, Vasyl & Hamid, Alain & Okhrin, Yarema, 2014. "The empirical similarity approach for volatility prediction," Journal of Banking & Finance, Elsevier, vol. 40(C), pages 321-329.
    10. Todd Guilfoos & Andreas Pape, 2016. "Predicting human cooperation in the Prisoner’s Dilemma using case-based decision theory," Theory and Decision, Springer, vol. 80(1), pages 1-32, January.
    11. Gayer, Gabrielle, 2010. "Perception of probabilities in situations of risk: A case based approach," Games and Economic Behavior, Elsevier, vol. 68(1), pages 130-143, January.
    12. Robert F. Bordley, 2011. "Using Bayes' Rule to Update an Event's Probabilities Based on the Outcomes of Partially Similar Events," Decision Analysis, INFORMS, vol. 8(2), pages 117-127, June.
    13. Pape, Andreas & Kurtz, Kenneth, 2013. "Evaluating Case-based Decision Theory: Predicting Empirical Patterns of Human Classification Learning (Extensions)," MPRA Paper 45206, University Library of Munich, Germany.
    14. Guerdjikova, Ani, 2008. "Case-based learning with different similarity functions," Games and Economic Behavior, Elsevier, vol. 63(1), pages 107-132, May.
    15. Teitelbaum, Joshua C., 2013. "Asymmetric empirical similarity," Mathematical Social Sciences, Elsevier, vol. 66(3), pages 346-351.
    16. Pape, Andreas Duus & Kurtz, Kenneth J., 2013. "Evaluating case-based decision theory: Predicting empirical patterns of human classification learning," Games and Economic Behavior, Elsevier, vol. 82(C), pages 52-65.
    17. Minjie Huang & Shunan Zhao & Andreas Pape, 2023. "Estimating Case‐based Individual and Social Learning in Corporate Tax Avoidance," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 85(2), pages 403-434, April.

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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:

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

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