Bayes Solutions of Sequential Decision Problems

Abstract

The study of sequential decision functions was initiated by one of the authors in [1]. Making use of the ideas of this theory the authors succeeded in [4] in proving the optimum character of the sequential probability ratio test. In the present paper the authors continue the study of sequential decision functions, as follows: a) The proof of the optimum character of the sequential probability ratio test was based on a certain property of Bayes solutions for sequential decisions between two alternatives, the cost function being linear. This fundamental property, the convexity of certain important sets of a priori distributions, is proved in Theorem 3.9 in considerable generality. The number of possible decisions may be infinite. b) Theorem 3.10 and section 4 discuss tangents and boundary points of these sets of a priori distributions. (These results for finitely many alternatives were announced by one of us in an invited address at the Berkeley meeting of the Institute of Mathematical Statistics in June, 1948) c) Theorem 3.6 is an existence theorem for Bayes solutions. Theorem 3.7 gives a necessary and sufficient condition for a Bayes solution. These theorems generalize and follow the ideas of Lemma 1 of [4] d) Theorems 3.8 and 3.8.1 are continuity theorems for the average risk function. They generalize Lemma 3 in [4] e) Other theorems give recursion formulas and inequalities which govern Bayes solutions.