  We study asymptotically minimax predictive distributions in an infinite
sequence model. First, we discuss the connection between the prediction in the
infinite sequence model and the prediction in the function model. Second, we
construct an asymptotically minimax predictive distribution when the parameter
space is a known ellipsoid. We show that the Bayesian predictive distribution
based on the Gaussian prior distribution is asymptotically minimax in the
ellipsoid. Third, we construct an asymptotically minimax predictive
distribution for any Sobolev ellipsoid. We show that the Bayesian predictive
distribution based on the product of Stein's priors is asymptotically minimax
for any Sobolev ellipsoid. Finally, we present an efficient sampling method
from the proposed Bayesian predictive distribution.
