On Bayesian and Decision-Theoretic Approaches to Statistical Prediction

Abstract

Let Y and Z be two random vectors with joint density f(y, z|θ), where θ∈Θ is an unknown parameter vector, and consider predicting Z based on y, the observed value of Y. We investigate Bayesian and decision-theoretic approaches to this problem, taking into account the loss function and the prior distribution of θ. Exploring connections between statistical prediction and decision theory, we find that a prediction problem can be reduced to a standard decision theory problem if the induced loss function is allowed to depend on the observed data y in addition to the unknown parameter θ and the decision d. In general, the predictive posterior density f(z|y) may not contain all information necessary for obtaining optimum predictions, but the posterior density f(θ|y) is adequate for that purpose. Some admissibility results are also discussed.