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Inductive Inference: An Axiomatic Approach

Author

Listed:
  • Itzhak Gilboa

    (TAU - Tel Aviv University)

  • David Schmeidler

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.

Suggested Citation

  • Itzhak Gilboa & David Schmeidler, 2003. "Inductive Inference: An Axiomatic Approach," Post-Print hal-00481297, HAL.
    • Handle: RePEc:hal:journl:hal-00481297
    DOI: 10.1111/1468-0262.00388
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    More about this item

    Keywords

    case-based reasoning; case-based decision theory; prediction; maximum likelihood; kernel functions; kernel classification;
    All these keywords.

    JEL classification:

    • C1 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General
    • D8 - Microeconomics - - Information, Knowledge, and Uncertainty

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