De Finetti's Theorem

Abstract

MOTIVATION We are finally back to discussing probabilities. To remind you where we left off, we had three definitions of probabilities. One, based on the principle of indifference, didn't bring us very far. The second, based on frequentism, was also limited. Subjective probabilities, by contrast, seemed very general and very flexible. But they are so general and so flexible that one might wonder if they can perhaps explain everything, accommodate everything, and eventually become useless. In order for the concept to be useful, subjective probabilities need a bit more discipline. Some rules that would tell us when our subjective probabilities are given by a probability measure p and when, by q . What is the difference, indeed, between such subjective probabilities? Is it possible that the difference between two such measures is not observable? And is it the case that everything can be explained by subjective probabilities? If so, what will the logical positivists say? Driven by such concerns, Ramsey (1931) suggested that subjective probabilities be derived from observable choice behavior. If subjective probabilities are computed from one's willingness to bet, they will have an observable meaning. Hopefully, we will be able to prove that two different probability measures are reflected in different modes of behavior, and can thus be told apart based on observations. Similarly, one should hope that certain modes of behavior will not be consistent with subjective probabilities, rendering the concept meaningful.