On recursive predictive distributions

A Bayesian framework is attractive in the context of prediction, but the need for Monte Carlo methods to compute the Bayesian predictive distribution precludes a fast recursive update in the context of streaming data. This paper demonstrates that the Bayesian predictive distribution update can be described through a bivariate copula density, making it unnecessary to pass through the posterior to update the predictive, opening the door for Bayesian online prediction. We show that, in standard models, the Bayesian predictive update corresponds to familiar choices of copula density. In nonparametric problems, it may not be possible to derive the appropriate copula, but our new perspective suggests a fast recursive approximation to the predictive density, in the spirit of Newton's predictive recursion algorithm for the Dirichlet process mixture posterior. Consistency of this new recursive algorithm is shown, and numerical examples demonstrate its quality performance in finite-samples compared to fully Bayesian and kernel methods.