Cognitive Foundations of Inductive Inference and Probability : An Axiomatic Approach ¤

We suggest an axiomatic approach to the way in which past cases, or observations, are or should be used for making predictions and for learning. In our model, a predictor is asked to rank eventualities based on possible memories. A \memory" consists of repetitions of past cases, and can be identi ̄ed with a vector, attaching a nonnegative integer (number of occurrences) to each case. Mild consistency requirements on these rankings imply that they have a numerical representation that is linear in the number of case repetitions. That is, there exists a matrix assigning numbers to eventuality-case pairs, such that, for every memory vector, multiplication of the matrix by the vector yields a numerical representation of the ordinal plausibility ranking given that memory. Interpreting this result for the ranking of theories or hypotheses, rather than of speci ̄c 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 their likelihood functions. As opposed to standard approaches, in our model there ¤We thank Didier Dubois and Peter Wakker for conversations that motivated this work; Daniel Lehman, Ariel Rubinstein, and Peyton Young for speci ̄c comments and examples; and Edi Karni and Gerda Kessler for many detailed comments. This paper contains most of the material in two earlier papers, \Inductive Inference: An Axiomatic Approach" and \Cognitive Foundations of Probability". This material is also presented as a Web paper at http://www.tau.ac.il/~igilboa/Inductive Inference/Index.html. yTel-Aviv University. igilboa@post.tau.ac.il zTel-Aviv University and The Ohio State University. schmeid@post.tau.ac.il