Inductive inference: An axiomatic approach

A predictor is asked to rank eventualities according to their plau-sibility, based on past cases. We assume that she can form a ranking given any memory that consists of Þnitely many past cases. Mild con-sistency requirements on these rankings imply that they have a numer-ical representation via a matrix assigning numbers to eventuality-case pairs, as follows. Given a memory, each eventuality is ranked accord-ing 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 lends to the plausibility of the eventual-ity. Special cases of this result may be viewed as axiomatizing kernel methods for estimation of densities and for classiÞcation problems. Interpreting the same result for rankings of theories or hypotheses, rather than of speciÞc eventualities, it is shown that one may ascribe to the predictor subjective conditional probabilities of cases given the-ories, such that her rankings of theories agree with rankings by the likelihood functions