Optimal estimates of predictive distributions

Approximations to Bayesian predictive density functions p ( y | x ) are sought with the use of the Kullback & Leibler (1951) directed measure of divergence. The approximations are constrained to lie in the same family of distributions as the underlying model p ( y |θ). The optimal approximations are found to coincide for the case of vague knowledge with some optimal estimates of p ( y |θ) derived by Murray (1977, 1979) and G. A. Bancroft.