The problem of sequential estimation of the mean, subject to the loss defin
ed as the sum of squared error loss and sampling costs, is considered withi
n the Bayesian framework. It is shown that the sequential procedure, as pro
posed by Chow and Yu (1981) in classical non-Bayesian sequential estimation
, is, in fact, asymptotically Bayes for a large class of prior distribution
s. The proposed procedure, without using any auxiliary data, is robust in t
he sense that it does not depend on the distribution of outcome variables a
nd the prior.