On Recursive Bayesian Predictive Distributions

P. Richard Hahn, Ryan Martin, Stephen G. Walker

Research output: Contribution to journalArticlepeer-review

21 Scopus citations


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 article 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. Supplementary materials for this article are available online.

Original languageEnglish (US)
Pages (from-to)1085-1093
Number of pages9
JournalJournal of the American Statistical Association
Issue number523
StatePublished - Jul 3 2018
Externally publishedYes


  • Copula
  • Density estimation
  • Nonparametric Bayes
  • Prediction
  • Recursive estimation

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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