On recursive Bayesian predictive distributions

TL;DR

This paper introduces a novel recursive Bayesian predictive update method using copulas, enabling fast online predictions without full posterior recalculations, and demonstrates its effectiveness through theoretical and numerical analyses.

Contribution

It presents a copula-based recursive Bayesian prediction method that simplifies updates and extends to nonparametric models, avoiding complex normalization computations.

Findings

The new method is consistent and theoretically sound.

Numerical experiments show competitive performance with existing methods.

The approach simplifies online Bayesian prediction in complex models.

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.