  The Bayesian predictive density has complex representation and does not
belong to any finite-dimensional statistical model except for in limited
situations. In this paper, we introduce its simple approximate representation
employing its projection onto a finite-dimensional exponential family. Its
theoretical properties are established parallelly to those of the Bayesian
predictive density when the model belongs to curved exponential families. It is
also demonstrated that the projection asymptotically coincides with the plugin
density with the posterior mean of the expectation parameter of the exponential
family, which we refer to as the Bayes extended estimator.
Information-geometric correspondence indicates that the Bayesian predictive
density can be represented as the posterior mean of the infinite-dimensional
exponential family. The Kullback--Leibler risk performance of the approximation
is demonstrated by numerical simulations and it indicates that the posterior
mean of the expectation parameter approaches the Bayesian predictive density as
the dimension of the exponential family increases. It also suggests that
approximation by projection onto an exponential family of reasonable size is
practically advantageous with respect to risk performance and computational
cost.
