The basic problem considered is where the observed data are all realizations of a random variable X and some probability statement about a future variable from the same distribution is desired. The paper examines such predictions with regard to a particular measure of prediction fit, the average Kullback-Leibler divergence between distributions. The notion of a bootstrap predictive distribution is introduced. This distribution is used in the place of a posterior distribution of the unknown parameter in Bayes' theorem. The ideas are illustrated with the Poisson and binomial distributions. The bootstrap method is superior to the estimative method substituting the estimates for the parameters. The same result is obtained in a large- sample framework, asymptotic in n, for natural exponential families.