Volume 31, Issue 2 p. 332-334
Article

A Note on the Posterior Mean of a Population Mean

W. A. Ericson

W. A. Ericson

University of Michigan

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First published: July 1969
Citations: 37

Summary

It is a well-known result, see for example Lindley (1965) and Raiffa and Schlaifer (1961), that if 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 , has the form {(n2) + m/v}/{(n2)+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 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.