  We consider Bayesian shrinkage predictions for the Normal regression problem
under the frequentist Kullback-Leibler risk function.
  Firstly, we consider the multivariate Normal model with an unknown mean and a
known covariance. While the unknown mean is fixed, the covariance of future
samples can be different from training samples. We show that the Bayesian
predictive distribution based on the uniform prior is dominated by that based
on a class of priors if the prior distributions for the covariance and future
covariance matrices are rotation invariant.
  Then, we consider a class of priors for the mean parameters depending on the
future covariance matrix. With such a prior, we can construct a Bayesian
predictive distribution dominating that based on the uniform prior.
  Lastly, applying this result to the prediction of response variables in the
Normal linear regression model, we show that there exists a Bayesian predictive
distribution dominating that based on the uniform prior. Minimaxity of these
Bayesian predictions follows from these results.
