A Cautionary Note for Bayesian Estimation of the Binomial Parameter n

Consider r independent realizations of a binomial random variable B(n, p). Given conventionally used priors for n and p it is shown that the tail weight of the marginal posterior density on n is never affected by the data. Consequently, the existence of important parameters associated with the posterior, such as the median and the mean, can be determined in advance by the prior, without regard to any finite amount of experimentation.