Experiment in the General Decision Problem

Abstract

We consider an experiment that conducts observations on an uncertain parameter. Experiments observing a parameter with a stochastic uncertainty have been studied exhaustively and their characteristics have been described by many authors [see, e.g., De Groot, M. (1974), Optimal Statistical Decisions (Russian translation)]. In this article, we assume that uncertainty is generated by a mechanism which is “random in the broad sense” [a term introduced by Kolmogorov, A.N. (1986), in Probability Theory and Mathematical Statistics (in Russia), pp. 467–471]. Ivanenko, V.I. and Labkovskii, V.A. [(1986), Dokladi Na Bolgarskata Akademiya Na Naukite SSSR 287(3), 564–567; (1990), Dokladi Na Bolgarskata Akademiya Na Naukite SSSR 310(5), 1059–1062] have proposed the apparatus of “statistical regularities” for the description of such randomness. Statistical regularity (SR) is a family of finitely additive probability distributions such that particular SRs correspond to total uncertainty, stochastic uncertainty, and other forms of uncertainty. The problem in which uncertainty is specified by SR is accordingly called General Decision Problem (GDP). We investigate informativeness and optimality of experiments in GDP.