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Inductive Inference: An Axiomatic Approach

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Author Info
Itzhak Gilboa (Tel Aviv University)
David Schmeidler (Tel Aviv University)

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Abstract

A predictor is asked to rank eventualities according to their plausibility, based on past cases. We assume that she can form a ranking given any memory that consists of finitely many past cases. Mild consistency requirements on these rankings imply that they have a numerical representation via a matrix assigning numbers to eventuality-case pairs, as follows. Given a memory, each eventuality is ranked according to the sum of the numbers in its row, over cases in memory. The number attached to an eventuality-case pair can be interpreted as the degree of support that the past case lends to the plausibility of the eventuality. Special instances of this result may be viewed as axiomatizing kernel methods for estimation of densities and for classification problems. Interpreting the same result for rankings of theories or hypotheses, rather than of specific eventualities, it is shown that one may ascribe to the predictor subjective conditional probabilities of cases given theories, such that her rankings of theories agree with rankings by the likelihood functions.

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File URL: http://cowles.econ.yale.edu/P/cd/d13a/d1339.pdf
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Paper provided by Cowles Foundation, Yale University in its series Cowles Foundation Discussion Papers with number 1339.

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Length: 45 pages
Date of creation: Dec 2001
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Handle: RePEc:cwl:cwldpp:1339

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Related research
Keywords: Inductive inference case-based reasoning case-based decision theory maximum likelihood

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Find related papers by JEL classification:
D80 - Microeconomics - - Information, Knowledge, and Uncertainty - - - General

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(explanations, Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.)

  1. Enriqueta Aragones & Itzhak Gilboa & Andrew Postlewaite & David Schmeidler, 2003. "Accuracy vs. Simplicity: A Complex Trade-Off," UFAE and IAE Working Papers 564.03, Unitat de Fonaments de l'Anàlisi Econòmica (UAB) and Institut d'Anàlisi Econòmica (CSIC). [Downloadable!]
    Other versions:
  2. Alan Beggs, 2005. "Learning in Bayesian Games with Binary Actions," Economics Series Working Papers 232, University of Oxford, Department of Economics. [Downloadable!]
  3. Massimiliano Amarante, 2006. "States, models and information: A reconsideration of Ellsberg's paradox," Discussion Papers 0506-18, Columbia University, Department of Economics. [Downloadable!]
  4. Massimiliano Amarante, 2006. "Analogical reasoning in decision processes," Discussion Papers 0506-17, Columbia University, Department of Economics. [Downloadable!]
  5. Itzhak Gilboa & Andrew Postlewaite & David Schmeidler, 2004. "Rationality of Belief Or Why Bayesianism is neither necessary nor sufficient for rationality," PIER Working Paper Archive 04-011, Penn Institute for Economic Research, Department of Economics, University of Pennsylvania. [Downloadable!]
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  6. Antoine Billot & Itzhak Gilboa & Dov Samet & David Schmeidler, 2003. "Probabilities: Frequencies Viewed in Perspective," Levine's Bibliography 666156000000000295, UCLA Department of Economics. [Downloadable!]
  7. Massimiliano Amarante, 2004. "States, models and unitary equivalence I: Representation theorems and analogical reasoning," Discussion Papers 0405-10, Columbia University, Department of Economics. [Downloadable!]
  8. Stefania D'Amico, 2005. "Density selection and combination under model ambiguity: an application to stock returns," Finance and Economics Discussion Series 2005-09, Board of Governors of the Federal Reserve System (U.S.). [Downloadable!]
  9. Antoine Billot & Itzhak Gilboa & Dov Samet & David Schmeidler, 2004. "Probabilities as Similarity-Weighted Frequencies," Levine's Bibliography 122247000000000696, UCLA Department of Economics. [Downloadable!]
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  10. David Schmeidler, 2000. "Utility in Case-Based Decision Theory," Working Papers 00-06, Ohio State University, Department of Economics. [Downloadable!]
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  11. Chambers, Christopher P. & Hayashi, Takashi, 2008. "Choice and individual welfare," Working Papers 1286, California Institute of Technology, Division of the Humanities and Social Sciences. [Downloadable!]
  12. Antoine Billot & Itzhak Gilboa & David Schmeidler, 2004. "An Axiomatization of an Exponential Similarity Function," Levine's Bibliography 122247000000000678, UCLA Department of Economics. [Downloadable!]
  13. Itzhak Gilboa & Andrew Postlewaite & David Schmeidler, 2004. "Rationality of Belief," Levine's Bibliography 122247000000000690, UCLA Department of Economics. [Downloadable!]
  14. Antoine Billot & Itzhak Gilboa & David Schmeidler, 2004. "Axiomatization of an Exponential Similarity Function," Cowles Foundation Discussion Papers 1485, Cowles Foundation, Yale University. [Downloadable!]
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