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On Recursive Bayesian Predictive Distributions

Author

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

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 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.

Suggested Citation

  • P. Richard Hahn & Ryan Martin & Stephen G. Walker, 2018. "On Recursive Bayesian Predictive Distributions," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 113(523), pages 1085-1093, July.
  • Handle: RePEc:taf:jnlasa:v:113:y:2018:i:523:p:1085-1093
    DOI: 10.1080/01621459.2017.1304219
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    Citations

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    Cited by:

    1. Ryan Martin, 2021. "A Survey of Nonparametric Mixing Density Estimation via the Predictive Recursion Algorithm," Sankhya B: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 83(1), pages 97-121, May.
    2. Lin, Edward M.H. & Sun, Edward W. & Yu, Min-Teh, 2020. "Behavioral data-driven analysis with Bayesian method for risk management of financial services," International Journal of Production Economics, Elsevier, vol. 228(C).
    3. Berti, Patrizia & Dreassi, Emanuela & Pratelli, Luca & Rigo, Pietro, 2021. "Asymptotics of certain conditionally identically distributed sequences," Statistics & Probability Letters, Elsevier, vol. 168(C).
    4. Patrizia Berti & Luca Pratelli & Pietro Rigo, 2021. "A Central Limit Theorem for Predictive Distributions," Mathematics, MDPI, vol. 9(24), pages 1-11, December.
    5. Jeong, Kuhwan & Chae, Minwoo & Kim, Yongdai, 2023. "Online learning for the Dirichlet process mixture model via weakly conjugate approximation," Computational Statistics & Data Analysis, Elsevier, vol. 179(C).
    6. Rodríguez, Carlos E. & Walker, Stephen G., 2021. "Copula Particle Filters," Computational Statistics & Data Analysis, Elsevier, vol. 161(C).
    7. Sandra Fortini & Sonia Petrone, 2020. "Quasi‐Bayes properties of a procedure for sequential learning in mixture models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 82(4), pages 1087-1114, September.

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