Carnap's inductive probabilities as a contribution to decision theory

Common probability theories only allow the deduction of probabilities by using previously known or presupposed probabilities. They do not, however, allow the derivation of probabilities from observed data alone. The question thus arises as to how probabilities in the empirical sciences, especially in medicine, may be arrived at. Carnap hoped to be able to answer this question byhis theory of inductive probabilities. In the first four sections of the present paper the above mentioned problem is discussed in general. After a short presentation of Carnap's theory it is then shown that this theory cannot claim validity for arbitrary random processes. It is suggested that the theory be only applied to binomial and multinomial experiments. By application of de Finetti's theorem Carnap's inductive probabilities are interpreted as consecutive probabilities of the Bayesian kind. Through the introduction of a new axiom the decision parameter λ can be determined even if no a priori knowledge is given. Finally, it is demonstrated that the fundamental problem of Wald's decision theory, i.e., the determination of a plausible criterion where no a priori knowledge is available, can be solved for the cases of binomial and multinomial experiments.