Sequential Bayes Estimation of the Difference Between Means

It is desired to estimate the difference between the means of two independent normal distributions as accurately as possible and in a sequential manner when the total number of observations is fixed. The problem is posed in a Bayesian framework with conjugate prior distributions and squared error-loss function. It is shown that the optimal sequential design depends on the ratio of the posterior variances of the two means. There exist constants (dependent on the prior parameters, the number of observations taken from each distribution, and the number of observations remaining) such that, when the above-mentioned ratio exceeds this constant, it is optimal to select the next observation from one distribution; otherwise it is optimal to select it from the other distribution.