Bayesian Analysis of Mixtures : Some results on Exact Estimability and Identification

Models frequently used for car insurance portfolios assume that the distribution of the expcted number of accident caused by a client is a mixture of Dirichlet Processes. Given that the number of clients is typically rather large, insurance companies may find it relevant to inquire wheter the posterior distribution (given the number of individual accidents) of the mixture, of the mixing parameters and the predictive distribution of a new observation, given the number of individual accidents, are consistently convergent. For most models of this rather general class, answering these questions requires unfathomable manipulations. In this paper, it is shown that answers are "easily" obtainable from general results, presented in the authors' monograph Elements of Bayesian Statistics. these answers rely on no specific assumption on the form of the distribution but only on simple conditions on latent process. Some problems of approximations are also discussed