A Note on the Posterior Mean of a Population Mean

Summary

It is a well-known result, see for example Lindley (1965) and Raiffa and Schlaifer (1961), that if x̄ is the mean of a sample of independent observations distributed N(μ, σ²) where σ² is known, and if μ has been assigned a normal prior distribution, N(m, v), then the posterior expectation of μ, given the sufficient statistic x̄, has the form {x̄(n/σ²) + m/v}/{(n/σ²)+1/v}, that is, has the intuitively appealing form of a weighted average of the prior mean and sample mean with weights inversely proportional to the prior variance and the conditional sampling variance of X̄ respectively. In this note we present an extremely simple generalization of this result which seems not to have been noted explicitly before and which holds for a variety of distributions other than the normal.