The De Finetti Transform

Abstract

Bayesian analysis of a problem requires that subjective information be introduced into the model. Sometimes it is more useful or convenient to bring that information to bear through the prior distribution; other times it is preferred to bring it to bear through the predictive distribution. This paper develops procedures for assessing exchangeable, proper, predictive distributions that, when desired, reflect “knowing little” about the model; i.e., exchangeable, maximum entropy, distributions. But what do such distributions imply about the model and the prior distribution? We invoke de Finetti’s theorem to define the “de Finetti transform”, which permits us, under many frequently occurring conditions (such as natural conjugacy), to find the unique sampling density (conditional on some indexing parameters), and the unique associated prior distribution for those parameters.