文档介绍：Cognitive Foundations of Inductive Inference
and Probability: An Axiomatic Approach¤
Itzhak Gilboayand David Schmeidlerz
March 2000
Abstract
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 eventuali-
ties based on possible memories. A \memory" consists of repetitions
of past cases, and can be identi¯ed with a vector, attaching a non-
negative integer (number of occurrences) to each case. Mild consis-
tency 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 the-
ories, 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¯ments and examples;
and Edi Karni and Gerda Kessler for many ments. 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 /~igilboa/Inductive Inference/.
yTel-Aviv University. ******@
zTel-Aviv University and The Ohio State University. ******@
1
is plete dichotomy between the set of objects that are ranked
(eventualities or theories) a