TY - JOUR

T1 - Inductive inference

T2 - An axiomatic approach

AU - Gilboa, Itzhak

AU - Schmeidler, David

PY - 2003

Y1 - 2003

N2 - 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.

AB - 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.

KW - Case-based decision theory

KW - Case-based reasoning

KW - Kernel classification

KW - Kernel functions

KW - Maximum likelihood

KW - Prediction

UR - http://www.scopus.com/inward/record.url?scp=0037273317&partnerID=8YFLogxK

U2 - 10.1111/1468-0262.00388

DO - 10.1111/1468-0262.00388

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AN - SCOPUS:0037273317

SN - 0012-9682

VL - 71

SP - 1

EP - 26

JO - Econometrica

JF - Econometrica

IS - 1

ER -