On the Greatest Class of Conjugate Priors and Sensitivity of Multivariate Normal Posterior Distributions

When samples are taken from a (multivariate) normal distribution then under suitable conditions we characterize the greatest class of priors such that the posterior distribution is also (multivariate) normal. In this case exact formulas are given showing how the mean and covariance matrix of the posterior normal distribution depend on the sample and on the a priori distribution; these formulas may be viewed as sensitivity of the posterior distribution to the prior and sample distribution.