Calibrating the prior distribution for a normal model with conjugate prior

For a normal model with a conjugate prior, we provide an in-depth examination of the effects of the hyperparameters on the long-run frequentist properties of posterior point and interval estimates. Under an assumed sampling model for the data-generating mechanism, we examine how hyperparameter values affect the mean-squared error (MSE) of posterior means and the true coverage of credible intervals. We develop two types of hyperparameter optimality. MSE optimal hyperparameters minimize the MSE of posterior point estimates. Credible interval optimal hyperparameters result in credible intervals that have a minimum length while still retaining nominal coverage. A poor choice of hyperparameters has a worse consequence on the credible interval coverage than on the MSE of posterior point estimates. We give an example to demonstrate how our results can be used to evaluate the potential consequences of hyperparameter choices.