  A Bayesian framework is attractive in the context of prediction, but a fast
recursive update of the predictive distribution has apparently been out of
reach, in part because Monte Carlo methods are generally used to compute the
predictive. This paper shows that online Bayesian prediction is possible by
characterizing the Bayesian predictive update in terms of a bivariate copula,
making it unnecessary to pass through the posterior to update the predictive.
In standard models, the Bayesian predictive update corresponds to familiar
choices of copula but, in nonparametric problems, the appropriate copula may
not have a closed-form expression. In such cases, our new perspective suggests
a fast recursive approximation to the predictive density, in the spirit of
Newton's predictive recursion algorithm, but without requiring evaluation of
normalizing constants. Consistency of the new algorithm is shown, and numerical
examples demonstrate its quality performance in finite-samples compared to
fully Bayesian and kernel methods.
