On recursive Bayesian predictive distributions

Abstract

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.

22 pages, 3 figures, 3 tables

References in corpus (2)

• Consistency of a recursive estimate of mixing distributions
• Stochastic Approximation and Newton's Estimate of a Mixing Distribution

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• On nonparametric estimation of a mixing density via the predictive recursion algorithm
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• Bayesian Quantile Estimation and Regression with Martingale Posteriors
• Revisiting consistency of a recursive estimator of mixing distributions
• Quasi-Bayes properties of a recursive procedure for mixtures