On the Posterior Mean and Variance of a Population Mean

This note shows that under a certain class of prior distributions of the vector, X = (X 1, …, XN ), of the unknown finite population values, the posterior mean and variance of the population mean, μ = Σ N 1 Xi /N, assume simple and intuitively appealing forms. For example, it is shown that the posterior mean of μ is a weighted average of the prior mean and the sample mean with weights inversely proportional to the prior variance of μ and the prior expected conditional sampling variance of the sample mean. The class of prior distributions of X is given by taking the Xi 's to be independent identically distributed conditional on some parameter θ, with conditional density in a particular exponential family, and assigning 8 a natural conjugate distribution. Similar properties are shown to hold for μ(θ) ≡ E(X|θ).