Bayesian Aspects in the Theory of Comparison of Statistical Experiments

Abstract

The purpose of this paper is to present some aspects of the theory of comparison of experiments where in an efficient way the Bayesian concept of apriori knowledge can be applied in order to enrich the motivation and the understanding of the mathematical analysis involved. There are various kinds of comparisons of experiments based on orderings of decision functions and their risks. Of particular interest in the applications are the Bayesian orderings introduced by De Groot and elaborated by Feldman. See [4], [3] and also [10], [11]. In the following we shall adopt the comparison invented by Blackwell and generalized by LeCam. Although this comparison is rather strong it has proved to be an important tool in asymptotic decision theory. Basic knowledge of the Blackwell-LeCam theory can be obtained from the text books [7] and [13]. We recall a few key notions. An experiment is determined by three data: a list of possible outcomes (the sample space (E, O₂)), a collection of possible explaining theories (the parameter set I), and a correspondence which to every explaining theory associates the random mechanism governing the random outcome (a mapping i → P_i. from I into the set M ¹(E, O₂) of probability measures on (E,O₂)). We shall consider experiments ε = (E,O2, P_i:iεI) with fixed parameter set I. Since there is no explicit definition of the information contained in an experiment, we content ourselves with the comparison of information whenever two experiments ε and ℱ = (F, B,Q_i:i ε I)are given.