Optimal Bayes procedures for selecting the better of two Bernoulli populations

Abstract

Let Π 1 and Π 2 be two Bernoulli populations with probabilities of success p 1 and p 2, respectively. In a recent paper Bechhofer and Kulkarni (1982) proposed a class of closed adaptive sequential procedures for the problem of selecting the population with the larger success probability. The procedures in this class achieve the same probability of correct selection, uniformly in ( p 1, p 2), as does the single stage procedure of Sobel and Huyett (1957) which takes exactly n observations from each population. This paper considers the problem of selecting an optimal procedure within this class which minimizes the expected total number of observations under the assumption that p 1 and p 2 have independent prior Beta distributions. Recursion equations are derived which can be used to determine such an optimal procedure. Empirical procedures are described which are shown numerically to perform very well when compared with the optimal procedures. The class of empirical procedures for symmetric Beta priors is conjectured to be minimax within the class of procedures considered by Bechhofer and Kulkarni; strong arguments are presented to support this conjecture. The empirical procedure using uniform prior distributions for p 1 and p 2 is shown to have very desirable properties. The performance characteristics of this procedure are studied in detail, and based on these it is proposed as an attractive procedure to use in practice.